Gemara

We offer Musaf of Shabbos before Musaf of Rosh Chodesh, for whatever is more Tadir (frequent) comes first.

Pesachim 115b (Beraisa): "Oni" is written without a Vov, so we can read it 'Ani' (a poor person). Just like an Ani has [only] a partial loaf, also here we eat from a piece.

Berachos 39b (Rav Papa): On Pesach (Seder night), all agree that one puts the piece in the loaf, to fulfill "Lechem Oni."

(R. Aba): On Shabbos one must recite the blessing on two loaves, to fulfill "Laktu Lechem Mishneh."

Rishonim

Rif (Pesachim 25b): On Pesach, we bless on one [whole] loaf and a piece. On other Yamim Tovim, we bless on two whole loaves, like on Shabbos. R. Aba taught that one must do so on Shabbos. Rabanan require two full loaves on Yom Tov, just like on Shabbos. The primary Chiyuv on Shabbos is because manna did not fall on Shabbos, rather, a double portion fell on Erev Shabbos. The same applies to Yom Tov! "Lechem Oni" detracts from one of them, i.e. we use a piece in place of a whole loaf. Therefore, one blesses on a whole loaf and a piece. He puts the piece in the loaf and cuts, to fulfill Lechem Oni, like Rav Papa taught.

Rosh (Pesachim 10:30): Some support him (the Rif) from the Mishnah "they brought in front of him Matzah", which connotes only one whole loaf. I disagree. Why is Pesach different than all other Yamim Tovim regarding Lechem Mishneh? "Lechem Oni" teaches about the bread for Mitzvas Achilas Matzah, but surely ha'Motzi requires two whole loaves, like other Yamim Tovim! A support is that if one has only a k'Zayis of Lechem Oni, he eats it at the end, and blesses ha'Motzi on Matzah Ashirah (kneaded with things other than water), surely he needs two whole loaves! The same applies if he has Lechem Oni to eat at the beginning. Why shouldn't he bless ha'Motzi on the whole loaf?! Therefore, the custom is to make three Matzos. In Ashkenaz and France, the custom is to make [them] from one Isaron (the volume of 43.2 eggs), to commemorate Lachmei (loaves brought with a Korban) Todah. One who leaves jail brings Lachmei Todah, which come from three kinds of Matzah. He breaks the middle Matzah in half, and puts the half between the two whole ones. He blesses ha'Motzi on the first Matzah, and Al Achilas Matzah on the piece, and eats together a k'Zayis from each. He wraps (the Maror) in the bottom Matzah, like Hillel. Since it was enacted three, we do a Mitzvah with each. One who wants to fulfill both opinions blesses ha'Motzi and Al Achilas Matzah and cuts from the whole Matzah and the piece together.

Gra (473:11): The manna did not fall on Yom Tov. A weekday must prepare for Yom Tov (Beitzah 2b). The Rif, Rambam, and others hold that one whole loaf and a piece suffice. It is not normal for an Oni to have two whole loaves and a piece.

Rambam (Hilchos Chametz 8:8): Nowadays that there is no Korban, after blessing ha'Motzi, he blesses Al Achilas Matzah.

Poskim

Shulchan Aruch (OC 475:1): One takes the three Matzos in the order he placed them, i.e. the broken is between the two whole ones, holds them in his hand, blesses ha'Motzi and Al Achilas Matzah, then cuts from the top whole Matzah and the piece, from both of them together.

Beis Yosef (OC 475 DH u'Mah she'Chasav she'Yesh): Also the Mordechai and Hagahos Maimoniyos in the name of Semag say so. The Rashba says that one takes a whole Matzah, puts a piece on it and blesses ha'Motzi and Al Achilas Matzah on the piece, due to "Lechem Oni.' He does not need three loaves. Even though we say "he puts the piece in the loaf", as long as the small one is in the big one, this is called "inside". The Tur says that the Rif holds like this, that two Matzos suffice. The Rosh disagrees, and this is the custom.

Taz (1): He holds them together because some rule that one blesses ha'Motzi on the whole loaf, and Al Achilas Matzah on the piece, which is Lechem Oni. Some say oppositely. Therefore, he holds both in his hand, blesses both Berachos, and then cuts from the whole loaf. The Tur says that one takes the plate. I.e. one does not take the whole Matzah in his hand. It must be with the third Matzah to fulfill Lechem Mishneh, like on every Yom Tov. Rather, they are on the plate, and he brings close to himself or takes all three Matzos in his hand and puts his hand on them, and after [blessing] he takes out the whole one, cuts, and then cuts from the piece. Maharshal says that he blesses ha'Motzi and breaks the whole loaf, but does not totally cut it. He holds the second and blesses Al Achilas Matzah, and breaks from both of them. Why does he break it without totally breaking it? Also, why does he divide the Berachos, i.e. ha'Motzi on the whole loaf and Al Achilas Matzah on the piece? He holds like one reasoning, but the Tur connotes that one should fulfill both opinions and say both Berachos one right after the other while they are on the plate, and then cut them. Also in Siman 486 the Tur said to cut them together.

Kaf ha'Chayim (5): The Ari Zal and Acharonim hold like this. (Ha'Motzi applies to the whole loaf, and Al Achilas Matzah to the piece.)

Magen Avraham (2): He holds all three together, to have Lechem Mishneh, and the piece for Lechem Oni.

Magen Avraham (3): This is even if it is on Shabbos [even though normally, on Shabbos night we cut from the bottom loaf].

Mishnah Berurah (2,3): After blessing ha'Motzi, he drops the bottom Matzah, and blesses Al Achilas Matzah on the top Matzah and the piece. He says both Berachos before breaking them. There should not be no Hefsek (interruption) between the first breaking and eating.

Kaf ha'Chayim (6): The Shlah says that this fulfills all opinions.

Kaf ha'Chayim (7): The Pri Chodosh says that we bless ha'Motzi first, for it is Tadir. One must intend to fulfill the Torah Mitzvah.

Bach (483:2 DH v'Ta'amo): Ba'al ha'Itur says that if one has no wine, when saying Kidush on the Matzah, he says Al Achilas Matzah right after ha'Motzi. He holds that the primary Mitzvah is Al Achilas Matzah, just we bless ha'Motzi first because it is Tadir. This is the reason for those who put the piece on the whole loaf.

Shulchan Aruch (ibid.): He eats them reclining together, a k'Zayis from each.

Taz (2): The two Berachos pertain to each other. Even if he had only one [Matzah, he would say both]. Since we do not do Mitzvos in bundles, he takes for each Berachah one Matzah. Therefore, he must eat them together, as if they were literally one Matzah. The Maharal asked that whatever cannot be done one after the other, cannot be done even together (Kidushin 50b). What is his question? Here, it is as if both are one body. This does not apply in Kidushin (e.g. being Mekadesh two sisters). This answers also the Bach's question, why we need a k'Zayis for ha'Motzi. One blesses ha'Motzi even on less than a k'Zayis (210:1)! This is not difficult. Here, the two Berachos take effect on two Mitzvos. There is no division, rather, both are for one eating. If so, why do we need two k'Zeisim? It should suffice to eat one k'Zayis from both of them! He does so in order to be Yotzei both of them, and ha'Motzi and Al Achilas Matzah apply to each of them. Therefore, he must eat a k'Zayis from each. However, this is difficult, for the Tur first said to bless ha'Motzi on the whole loaf, and Al Achilas Matzah on the piece, and even so he said to eat a k'Zayis from each! This is not difficult also for that opinion. The solution is that whatever he eats at the time, both Berachos should take effect on it together. If he eats a k'Zayis from one and less than k'Zayis from the other, the eating is uneven. At first he fulfills two Mitzvos, and at the end one Mitzvah.

Magen Avraham (4): Darchei Moshe brings from Terumas ha'Deshen (139) that one must eat a k'Zayis together and swallow. The Beis Yosef brings so from Tosfos. This connotes that it suffices to swallow one k'Zayis at once. The Mordechai asked "one cannot swallow all three (Pesach, Matzah and Maror) together? One can swallow only k'Beitzah at once, i.e. two k'Zeisim! He answered that when he swallows, they are minced, so they all fit. This shows that one may not swallow the k'Zayis of Matzah bit by bit, for if so, he had no question! Rather, Hillel obligates swallowing them together, and we hold similarly about the k'Zayis of Matzah or Maror." I.e. it suffices to swallow one k'Zayis at once. The Shulchan Aruch says to eat them together, i.e. put them in the mouth at once. Maharil permits eating slowly, as long as he does not delay Kdei Achilas Pras while eating [the k'Zayis]. Sa'if 6 connotes that this is b'Di'eved, but l'Chatchilah one must eat them together. Chulin 103b connotes like this. For Maror, one may rely on Maharil if it is hard for him to swallow it together.

Gra (DH va'Yvarech): Tosfos (39b DH ha'Kol) says we cannot make both Berachos on the piece, for we do not do Mitzvos in bundles. R. Menachem made both Berachos on the piece. This is not called bundles, just like we bless Borei Peri ha'Gafen and Mekadesh ha'Shabbos on the cup of Kidush. If one has no vegetable other than Maror, he blesses Borei Peri ha'Adamah and Al Achilas Maror the first time he eats it. The Ri did like the Shulchan Aruch says, to avoid the argument.

Gra (473:11): Doing Mitzvos in bundles does not apply to Berachos.

Gra (DH b'YAchad): If he eats of ha'Motzi first, he is Yotzei with it, but we require a piece [for Lechem Oni]!

Mishnah Berurah (8): He eats both together, lest he interrupt between a Berachah and eating the Matzah corresponding to that Berachah.

Mishnah Berurah (9): It suffices to put them in his mouth at once. He need not swallow them at once. He should swallow about a k'Zayis at once, and the rest at once. If he swallowed them slowly, b'Di'eved, he was Yotzei if he did not delay Kdei Achilas Pras.

Bi'ur Halachah (DH k'Zayis): The Gemara mentions only one k'Zayis. The only Poskim who mention two k'Zeisim are the Rosh and Mordechai. This requires investigation.

Shulchan Aruch (ibid.): If he cannot eat two k'Zeisim together, he eats first that of ha'Motzi, and afterwards of Al Achilas Matzah.

Taz (3): Also the Tur (486) says to eat of ha'Motzi first. This is like the first opinion in the Tur, that he blesses ha'Motzi on the whole loaf, and Al Achilas Matzah on the piece, so we know which is ha'Motzi. Therefore it is first, if one cannot eat them together. However, he wrote afterwards to bless ha'Motzi and Al Achilas Matzah on both of them. If so, we cannot distinguish which is [of Al Achilas] Matzah and which is ha'Motzi! Why did the Shulchan Aruch and Tur write this according to this opinion? It allows eating either first! One can say that since the whole loaf is in front of him first, he eats from it first, lest he pass over Mitzvos. If so, if he puts down the piece [that he broke] from the whole loaf, and the piece [of the middle Matzah] is still in his hand, he should not put it down, for this is passing over Mitzvos.

Gra (DH v'Im): If one ate the Matzah half at a time, he was Yotzei b'Di'eved (Pesachim 114b).

Mishnah Berurah (10): B'Di'eved, eating the top whole Matzah first is not a Hefsek, since it is needed for the k'Zayis, i.e. for Birkas ha'Motzi that one must bless also on the piece.

Mishnah Berurah (11): B'Di'eved one was Yotzei whether he ate from the whole Matzah or the piece.