Gemara

18b (Beraisa): A baker may not fill a flask with water and put it in an oven just before Shabbos.

Rejection: This is even like Beis Hillel. This is a decree, lest one stoke the coals [on Shabbos, to hasten the cooking].

Question: If so, they should forbid leaving dye in a pot!

Answer (Shmuel): They permit when the pot is off the fire.

Question: Still, it should be forbidden, lest he stir the pot!

Answer: The case is, the pot is plastered shut.

34a (Rava): After dark, Hatmanah (wrapping a food to keep it hot) is forbidden [even] in something that does not add heat. This is a decree lest one heat the food before wrapping it.

39a (Mishnah): Any food that that was soaked in hot water before Shabbos, one may soak it in hot water on Shabbos.

145b: An example [of a food that is soaked and soaked again] is R. Aba's chicken. [He does so for the sake of Refu'ah.]

Beitzah 34a (Beraisa): If one brought the fire, one brought the wood, one set the pot over the fire, one brought the water, one put spices in it, and one stirred, all of them are liable.

Rishonim

Rambam (Hilchos Shabbos 9:3): If one cooks on the fire something fully cooked or something that does not need to cook at all, he is exempt.

Magid Mishneh: He learns from the Mishnah "any food that that was soaked in hot water before Shabbos, one may soak it in hot water on Shabbos." This means that Bishul does not apply. However, mid'Rabanan one may not put it on the fire. We learn from Chazarah of Mitztamek v'Ra Lo.

Rosh (3:11): Rashi (34a) explained that we decree lest one heat the food on the fire before wrapping it, and he cooks on Shabbos. Also on 18b, Rashi said that if he stirs a pot that was already cooked, this is Bishul mid'Rabanan. This shows that even if a Tavshil was cooked, if it cooled off, Bishul applies, since there is liquid. Any food that that was soaked in hot water before Shabbos, one may soak it in hot water on Shabbos. This refers to a dry food, e.g. R. Aba's chicken. I bring a proof for Rashi from the Beraisa in Beitzah. Both the one who put water in the pot and the one who stirred are liable. However, perhaps one is liable for hastening Bishul until the food is k'Ma'achal Ben Drusai, and after that Ein Bishul Achar Bishul. However, if it cooled off and one boiled it, he is liable, like Rashi explained the decree lest one reheat. We decree lest one stir a pot with dye, even if it is still boiling. Perhaps dye always needs to be cooked. Alternatively, there the problem is not cooking, rather, dying the wool.

Rashi (Shabbos 34a DH mishe'Chashechah): We forbid Hatmanah [after dark], lest one see that the food cooled off, and he will boil it and cook it before wrapping it.

Gra (OC 318 DH Tavshil): This is even if it was already fully cooked, like Tosfos says (47b DH ba'Meh), that Stam pots [are already cooked Bein ha'Shemashos].

Beis Yosef (OC 318 DH v'Nir'eh): The Tur should have ruled like the Rosh said, that if it reached k'Ma'achal Ben Drusai and cooled off, one who boils it is liable. The Tur wrote like this in Kitzur Piskei haRosh. Here he says that Bishul applies even while it is boiling! It seems that Kitzur Piskei ha'Rosh is to explain the Rosh. Here the Tur comes to rule in practice, so he did not want to rely on the Rosh, for he has no proof. The Beraisa in Beitzah says that Stam that they are liable. It did not distinguish whether or not it reached k'Ma'achal Ben Drusai. The Tur does not rely on the weak answers given to the question from stirring on 18b. Rather, Bishul applies even after k'Ma'achal Ben Drusai, even when it is boiling. We must say that Rashi discussed after it cooled off, for then Bishul applies even if it cooked totally. A Stam pot wrapped in something that does not add heat is fully cooked, and one is liable for cooking it if it cooled off. Do not say that the Tur rules like the Rosh, and the Rosh distinguishes between before and after k'Ma'achal Ben Drusai only regarding Chiyuv Chatas, but he forbids even after k'Ma'achal Ben Drusai. If Bishul applies, one is liable for it after k'Ma'achal Ben Drusai. If Bishul does not apply, why is it forbidden? Also, the Rosh wrote his reason that after it reached k'Ma'achal Ben Drusai, if it cooled off and he boiled it he is liable, like Rashi said. He did not distinguish whether or not it was fully cooked. What is the Tur's source to distinguish? Therefore, what I said above is primary. I say that Yad Soledes Bo is called boiling.

Beis Yosef (DH v'Ha): Semak says that all agree that Bishul applies to something fully cooked that cooled off if it has liquid, like Rashi says on 34a. R. Yerucham says so if it is Mitztamek v'Yafeh Lo, therefore one may not put it close to a fire where it is Yad Soledes Bo. He says that R. Yonah permits Chazarah if food reached k'Ma'achal Ben Drusai, but not if it has liquid and cooled off and it is Mitztamek v'Yafeh Lo. This is absolute Bishul! This implies that Semichah is permitted if it is Mitztamek v'Ra Lo. He also says that even if it ceased boiling a little, if it did not cool off one may return it. The Rambam connotes that if it was not fully cooked, even if it reached k'Ma'achal Ben Drusai, Bishul applies and he is liable. The Magid Mishneh says that the Rashba and some Meforshim say that there is no Torah Chiyuv after k'Ma'achal Ben Drusai. If so, one may soak it in hot water, since there is no Chiyuv on the fire. They did not distinguish dry foods from those with liquid. Also on Halachah 22:4, the Magid Mishneh wrote that the Rashba permits Semichah, in a place that is Yad Soledes Bo, of something cooked before Shabbos that cooled off. He did not stipulate that it must be dry. It seems that the Ran agrees.

Kaf ha'Chayim (53): The Beis Yosef holds that if it is Yad Soledes Bo, if one boils it he is exempt. If it was not Yad Soledes Bo, and he made it Yad Soledes Bo, he is liable.

Poskim

Shulchan Aruch (OC 318:4): If a Tavshil was fully cooked, Bishul applies if it cooled off.

Magen Avraham (11): Whatever is forbidden due to Bishul, it is forbidden even to put on the oven before it is heated, like I wrote above (253:43).

Eshel Avraham: I.e. even if the Nochri will heat it later, it is forbidden.

Note: If one puts a raw food on a cold hot-plate, and later a Shabbos clock will turn it on, the Chazon Ish (Mo'ed 38:2,3) says that this is Gerama. It is permitted only to avoid a loss. Also, the laws of Shehiyah apply also in such a case. Shmiras Shabbos k'Hilchasah (1:26) permits telling a Nochri to put food on such a hot-plate only if it was cooked, and even if it cooled off.

Magen Avraham (12): Whatever is Yad Soledes Bo is considered boiling.

Mishnah Berurah (23): The Torah forbids putting this Tavshil in a place that is Yad Soledes Bo. One may be lenient to mix it into a boiling Tavshil in a Kli Sheni.

Mishnah Berurah (24): The Rema (Sa'if 15) says that as long as it did not totally cool off, the custom is to be lenient to say that Bishul does not apply.

Kaf ha'Chayim (50): Whenever Bishul applies, it is forbidden even if he cooked through a Nochri.

Kaf ha'Chayim (51): The Rambam and Ran are lenient even if it has liquid. The Rema says that the custom is to be lenient if it did not cool off totally. Where there is no custom, one should be stringent like the Shulchan Aruch. However, the Birkei Yosef says that one may be lenient through a Nochri, since one opinion permits through a Yisrael.

Rema: Some say that this is only if it is Mitztamek v'Yafeh Lo.

Gra (DH v'Yesh Omrim): This is like Sa'if 8 brings from R. Yerucham. However, above (257:7) the Shulchan Aruch forbids Hatmanah in every case. Perhaps R. Yerucham would permit also Hatmanah (of Mitztamek v'Ra Lo).

Damesek Eliezer: We forbid Hatmanah lest one reheat the food before wrapping it. If R. Yerucham exempts Bishul Achar Bishul of Mitztamek v'Ra Lo, he could permit also Hatmanah. If so, the Rema should have brought above that some permit Hatmanah of Mitztamek v'Ra Lo!

Mishnah Berurah (25): The Bach holds like the first opinion, that does not distinguish Mitztamek v'Ra Lo from Mitztamek v'Yafeh Lo. The Gra connotes similarly.

Kaf ha'Chayim (56): Some say that the Shulchan Aruch agrees to the Rema, just we are not experts about what is Mitztamek v'Ra Lo, so we are stringent.

Shulchan Aruch (ibid.): If it was not fully cooked, even if it reached k'Ma'achal Ben Drusai, Bishul applies, even when it is boiling.

Gra (DH v'Afilu): This is unlike the Rosh (Damesek Eliezer - rather, like the) Tur and Rambam.

Mishnah Berurah (27): One may not finish the cooking through a Nochri. B'Di'eved, we do not forbid the food. We rely on the Poskim who hold that since it was cooked k'Ma'achal Ben Drusai, Bishul does not apply any more.

Kaf ha'Chayim (60): The Radvaz (1:213) says that if something is eaten raw, one may not cook it, but one is exempt for it, just like something k'Ma'achal Ben Drusai. This connotes that one is liable according to the opinion that obligates for cooking past k'Ma'achal Ben Drusai. The Shulchan Aruch (254:4) connotes like this.

Shulchan Aruch (ibid.): There is Bishul Achar Bishul only in a Tavshil with liquid.

Mishnah Berurah (29): When a liquid cools off, its initial Bishul is Batel.

Kaf ha'Chayim (62): "It has liquid" means that it is mostly liquid.

Shulchan Aruch (ibid.): If a dry food was already cooked, one may soak it in hot water on Shabbos.

Magen Avraham (14): One may soak it even in a Kli Rishon, for Ein Bishul Achar Bishul, like Tosfos, the Mordechai and Ran, unlike the Bach, who is stringent. The Shulchan Aruch connotes that if it merely soaked before Shabbos, but was not cooked, one may not soak it on Shabbos. It is good to be stringent if the water is Yad Soledes Bo.

Mishnah Berurah (31): If it merely soaked before Shabbos, but was not cooked, one may not soak it on Shabbos in water that is Yad Soledes Bo. Even though there is no Tosefes Bishul (i.e. it is not a more potent form of cooking - PF), it does cook a little more. Some are lenient. The Pri Megadim says that they are lenient about something that cooks easily even in a Kli Rishon off the fire. If not, all forbid. One may soak it in a Kli Sheni.

Mishnah Berurah (32): "Dry" means that the liquid was removed.

Shulchan Aruch (ibid.): If a dry food was not cooked before Shabbos, one may not soak it in hot water on Shabbos, but one may rinse it in hot water on Shabbos, except for old salted fish or tuna fish, for rinsing them is their final Melachah.

Magen Avraham (15): One may rinse from a Kli Sheni, but one may not soak it even in a Kli Sheni, for this looks like Bishul. This is unlike spices, which are to sweeten the Tavshil (one does not intend to eat the spices themselves). I say that an onion is like spices. It is for taste.