Gemara

(Rav Yehudah): If a Yisrael left meat on coals, and a Nochri flipped (the meat), it is permitted.

Question: What is the case?

If the meat would have cooked even had the Nochri not flipped, obviously it is permitted!

Answer #1: It would not have cooked had the Nochri not flipped.

Rejection: If so, the Nochri caused it to be cooked. It should be forbidden!

Answer #2: Had the Nochri not flipped, it would have cooked in two hours. Because he flipped, it cooked in one hour. One might have thought that because he hastened the cooking, it is considered Bishul Akum. Rav Yehudah teaches that this is not so.

Question: Rav Asi taught that anything cooked (by a Yisrael) k'Ma'achal Ben Drusa (like the food of a thief who used to eat food one third or one half cooked) is not considered Bishul Akum if a Nochri finishes cooking it.

Inference: If something was cooked less than this amount, if a Nochri finishes cooking it, it is Bishul Akum!

Answer: That is when the Yisrael (took it off the fire and) put it in a basket after partially cooking it. (In Rav Yehudah's case, the Yisrael left it on the fire. It would have finished cooking by itself.)

Support (Beraisa): A Yisrael may leave meat on coals and allow a Nochri to turn it over until he (the Yisrael) returns from the Beis ha'Keneses or Beis Medrash. A Yisraelis may leave a pot on a stove and allow a Nochris to mix it until she returns from the bathhouse or Beis ha'Keneses.

Question: If a Nochri left meat on coals and a Yisrael flipped, what is the law?

Answer (Rav Nachman bar Yitzchak): We permit when a Yisrael left meat on coals and a Nochri flipped (even though the Nochri finished the cooking), and all the more so it is permitted when the Yisrael finishes the cooking!

(Rabah bar bar Chanah): Whether a Yisrael left meat and a Nochri flipped, or vice-versa, it is permitted. It is forbidden only if the Nochri began and finished the cooking.

(Ravina): The Halachah is: if a Nochri lit an oven and a Yisrael put the bread in, or vice-versa, or if a Nochri did both and a Yisrael stoked the coals (this increases the heat), the bread is permitted.

Rishonim

Rambam (Hilchos Ma'achalos Asuros 17:16): If a Yisrael did some of the cooking at the beginning or end, it is permitted. Therefore, if a Nochri left meat or a pit on a fire, and a Yisrael turned over the meat or stirred the pot, or a Yisrael left it and a Nochri finished the cooking, it is permitted.

Rosh (2:32): Some forbid a food that a Nochri cooked to k'Ma'achal Ben Drusai and a Yisrael finished it. Since we permit when a Yisrael cooked to k'Ma'achal Ben Drusai and a Nochri finished it, this shows that k'Ma'achal Ben Drusai is considered cooked, and there is no cooking after cooking. This is wrong. We consider it like cooked to be lenient, but not to be stringent. We say that hastening cooking is insignificant when a Yisrael began the cooking and a Nochri hastened it, but it is significant to permit when a Nochri began the cooking and a Yisrael hastened it. Chachamim were lenient about Bishul Akum; we do not learn from a Kal va'Chomer.

Ran (15b DH Pas): If a Yisrael lit the oven and a Nochri baked the bread, it is permitted. Igniting the oven is a special Melachah for bread, but not for other cooked foods. The Sugya connotes like this. It discusses putting meat or pots on a fire, but connotes that just lighting the fire does not help. All the Rishonim say so. There is no Safek about this.

Poskim

Shulchan Aruch (YD 113:6): If a Yisrael did some of the cooking, whether at the beginning or at the end, it is permitted. Therefore, if a Nochri left meat or a pot on a fire, and a Yisrael turned over the meat or stirred the pot, or if a Yisrael stirred and a Nochri finished, it is permitted.

Beis Yosef (DH v'Lo): We conclude that it is forbidden only if the Nochri began and finished the cooking.

Beis Yosef (DH umi'Kol): When the Yisrael put (the food on the fire) and the Nochri flipped, we permit only if he merely hastened the cooking, but if it would not have cooked without his flipping it is forbidden. Why did the Rambam and Tur omit this? Perhaps the Tur relied on what he wrote afterwards about a Yisrael who put meat on Omemos coals (if it would not have cooked without the Nochri stoking, it is forbidden). Perhaps the Rambam held that this is obvious, so he did not need to write it.

Beis Yosef (DH v'Nir'eh): If a Yisrael left a spit with meat on it near the fire where it can roast, one may let a Nochri flip it so it will roast only if the Yisrael roasted it k'Ma'achal Ben Drusa. (If not,) if the Nochri would not flip it, the other side would not roast at all! Also, we must be concerned lest the Nochri remove it and return it, unknown to the Yisrael. This is unlike a pot that a Yisrael set down and a Nochri stirs it. There we are not concerned lest he remove it. Here, the spit often comes out of its place, and when the Nochri returns it, it is forbidden.

Rema: This is even if it would not have cooked without the Nochri's help.

Question (Taz 6): The Gemara said that if it would not have cooked had the Nochri not turned it over, it is forbidden! The Darchei Moshe brings this from the Beis Yosef! Rather, the Rema means that without the Nochri's help it would have cooked k'Ma'achal Ben Drusa, but not fully. In any case we hold like the opinion the Rema brings (Sa'if 7) that it suffices for the Yisrael to stoke or light the fire. It holds that when the Gemara says 'it is forbidden only if the Nochri began and finished the cooking', it retracts from what it said above (that if it would not have cooked had the Nochri not turned it over, it is forbidden). I say that one may be lenient only in the Yisrael's house, for the Tur brings an opinion that always permits then. In a Nochri's house, the Yisrael must put it were it can cook.

Shach (8): Also Isur v'Heter permits. It seems that he and the Rema hold that according to the Gemara, it is forbidden, just like stoking or adding wood. However, we hold that adding wood permits (regarding bread of Nochrim - 112:10). We rely on the difference on customs between Bnei Eretz Yisrael and Bnei Bavel. It is better that Yisrael be Shogeg than Mezid, like the Rosh and Tur say there. The same applies to Bishul Akum. Perhaps here the Rema discusses when the Yisrael put the coals and made the fire, like in Sa'if 7. However, the Ran gave reasons to distinguish other foods from bread.

Shulchan Aruch (7): Igniting the oven permits only regarding bread, but igniting the oven or lighting the fire does not matter regarding other cooked foods. Only putting the food on the fire matters. Therefore, if a Yisrael wants to cook in a pan in a Nochri's oven, he must put it in the oven in a place where it can get cooked.

Beis Yosef (DH v'Lo): We conclude that it is forbidden only if the Nochri began and finished the cooking. I say that 'flipping' refers to flipping the meat. The Rambam explains this way. He did not turn it so the other side faces the coals. If the Nochri removed it from the fire before it was k'Ma'achal Ben Drusa and returned it, it is forbidden! Rather, he shakes it and is Mone'a (prevents) the fire (from roasting part of it; Hagahos Tur ha'Shalem 15 brings a text 'Munach', i.e. it is on the fire the entire time), and through this the coals flared up and hastened the cooking. The Tur explains that he flipped (stoked) the coals. The Ran, Rashba, R. Yonah (Igeres ha'Teshuvah 38) and Rivash (514) say that igniting the oven permits only regarding bread, but not for other cooked foods. R. Yonah adds that a Nochri should not put a pan (with food that a Yisrael will eat) even if a Yisrael put wood in the oven, for this permits only bread, but not other cooked foods. The Yisrael must put the pan in the oven in a place where it can get cooked. It seems that they hold that all the more so, stoking the fire does not permit other foods. They must explain like the Rambam, that 'flipping' refers to the meat. However, we could say that lighting the fire does not permit meat because it is done before the meat is there, so it is not evident that he lit it to cook the meat. Perhaps he did so for warmth, or to cook something that (Bishul Akum does not apply to it, for it) does not come on kings' tables, or is eaten raw. Usually, one lights an oven only to bake. If one stokes coals while meat is over them, this hastens the cooking, and it is permitted even according to the Ran. This is difficult.

Rema: Some disagree and say that lighting the fire or stoking the coals helps for cooking, just like for bread. This is the custom. Even stoking without intent helps. Some say that even if the Yisrael did not stoke or throw in a chip of wood (to increase the fire), but the Nochri lit the Yisrael's fire (i.e. fuel), it is permitted.

Taz (8): Isur v'Heter says that stoking without intent does not help. Darchei Moshe says that even blowing the fire with his mouth helps.

Shach (9): The stoking must hasten the cooking.

Gra (18): Hastening cooking is significant, like we say about hastening Avodah (Shevuos 17b). We say that it is not significant when this is a leniency. The Rema says that it helps for the Yisrael to light the fire. Really, we permit only when the Yisrael finishes, but not when he just begins.