Gemara

(Rav Huna): If a non-Kohen flipped a fork, he is Chayav Misah b'Yedei Shamayim (because this is an Avodah).

Question: If the limbs were consumed only because he flipped them, obviously he is liable for Avodah! If the limbs would have been consumed in any case, he accomplished nothing. Why is he liable?

Answer: The case is, the limbs would have been consumed, but not as soon. Rav Huna teaches that hastening the Avodah is considered Avodah.

Beitzah 34a (Beraisa): If one brings the fire, and one brings the wood, and one sets the pot down, and one puts in the water, and one puts in the spices, and one stirs, all of them are liable (if there was a fire from the beginning).

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

This is a decree, lest one stoke the coals (on Shabbos to hasten the cooking).

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

Answer (Shmuel): It is permitted when the pot is off the fire.

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

Answer: The case is, the pot is plastered shut (it is an exertion to open it, before doing so he would remember that it is Shabbos).

Rishonim

Rambam (Hilchos Shabbos 9:3): If one cooks on fire something that is fully cooked, or need not be cooked, he is exempt.

Rosh (Shabbos 1:34): One may not put water in an oven just before Shabbos, lest it cool off towards the end of Shabbos, and he stoke the coals so it will cook immediately after dark. We decree about flax in a pot off the fire if it is not plastered, lest one stir it. Rashi explains that he would be liable for cooking while it is boiling, even though it is off the fire. All the more so one may not put a ladle into a pot on the fire.

Rosh (3:11): Rashi explained that we are concerned lest one heat up a food that cooled off, and be liable for cooking. On 18b, Rashi says that Bishul applies to something that was cooked, i.e. if it cooled off and has liquid. From his words (this is the Bach's text), the Heter to soak (on Shabbos) in hot water something that was cooked applies to something dry, e.g. the chicken of R. Aba (Shabbos 145b. It is soaked many days, for healing.) Beitzah 34a supports Rashi. If one brings the fire, and one brings the wood... all of them are liable. Even though the one who brings the water is liable for cooking, the one who stirs is liable. This shows that Bishul applies after Bishul, even if it was still boiling. However, one can reject this. Until it is k'Ma'achal ben Drusai (the way a certain thief used to eat, i.e. a half or a third cooked), one is liable for hastening the Bishul. After it reached k'Ma'achal ben Drusai, perhaps there is no Bishul after Bishul. Presumably, until it is k'Ma'achal ben Drusai, one is liable for hastening the Bishul. After it reached k'Ma'achal ben Drusai, if it cooled off and one heated it, he is liable, like Rashi said. We can say that we decree lest one stir wool, because it always needs Bishul. Alternatively, we did not ask due to Bishul, rather, due to dying the wool.

Kitzur Piskei Rosh (3:11): Before a food is k'Ma'achal ben Drusai, one is liable for hastening Bishul. After it reached this stage, as long as it is boiling, Bishul does not apply. If it cooled off and one heated it, he is liable, i.e. if it is a liquid. Bishul after Bishul does not apply to solids.

Ritva (17b DH v'Ha): Here we say that hastening Avodah is Avodah. If a Yisrael put meat on coals, and a Nochri turned it over it is not considered Bishul Nochri, even though he hastened the Bishul (Avodah Zarah 38a)! Even if Bishul Nochri were mid'Oraisa, we could say that the Torah is more stringent about Avodah. Really, here and for cooking on Shabbos, we follow Torah law (it is Bishul). We are lenient about Bishul Nochri, which is mid'Rabanan, also when a Nochri put the meat a Yisrael hastened the Bishul. We forbid only when a Nochri totally cooked it.

Poskim

Shulchan Aruch (OC 318:4): If a food is not fully cooked, even if it is k'Ma'achal ben Drusai, Bishul applies to it, even if it is still boiling.

Gra (13): This is unlike the Rosh, (Dameshek Eliezer - rather, like the) Tur and Rambam.

Mishnah Berurah (26): Bishul applies mid'Oraisa until it is fully cooked.

Mishnah Berurah (27): One may not even tell a Nochri to finish cooking it. B'Di'eved, one may rely on the lenient opinions.

Shulchan Aruch (18): If one took a boiling pan or pot off the fire, if the food was not fully cooked, one may not remove the food with a ladel, for this stirs and cooks.

Kaf ha'Chayim (171): If it was stirred once and this suffices to cook it, one is exempt for stirring more. In Siman 253, the Beis Yosef cites the Kolbo, who says that one who stirs a fully cooked food on the fire is liable.

Shulchan Aruch (ibid): If the food is fully cooked, it is permitted. One may not stir wool in a pot, even if it absorbed the dye.

Beis Yosef (DH Afilu): Bishul does not apply to what was already cooked. In Shabbos, we decree to forbid Hatmanah (wrapping a food to keep it warm) in Davar she'Eino Mosif Hevel (something that does not add heat), lest one heat the food. There, the decree is not lest he cook, rather, lest he stoke coals. Rashi concludes that one is liable for cooking something that was fully cooked only if it he heated it up after it cooled off. If it was (still) boiling, he would be exempt, for it reached k'Ma'achal ben Drusai. If so, Rashi should rule like the Rosh, that if it reached k'Ma'achal ben Drusai and cooled off and he heated it, he is liable, like the Remazim (the Tur's Kitzur Piskei Rosh, 3:11). Here, the Tur says that Bishul applies even while it is boiling! Perhaps the Tur does not write his own reasoning in the Remazim; he just abridges the Rosh. In the Arba'ah Turim, he teaches the Halachah, so he did not rely on the Rosh, for there is no proof for him. A Beraisa (Beitzah 34a) says Stam that all are liable; it does not distinguish whether or not the food reached k'Ma'achal ben Drusai. The Gemara asked why we do not decree lest he stir the pot, and gave a weak answer; we do not rely on it. Rather, Bishul applies even after food is k'Ma'achal ben Drusai, even if it is still boiling. Rashi said that we decree lest he heat something that was cooked and cooled off, i.e. after it was fully cooked. Therefore, he is liable only if it cooled off. If it was never fully cooked, he is liable even if it is still boiling. Also, one does Hatmanah in Davar she'Eino Mosif Hevel only if the food was fully cooked. Even so, one is liable if it cooled off and he heated it up. Do not say that the Tur rules like the Rosh. The Rosh distinguishes whether or not it reached k'Ma'achal ben Drusai only regarding Chiyuv Chatas, but one may not cook it more even after it reached k'Ma'achal ben Drusai.

Gra (51): The Heter to return a fully cooked food to the fire shows that Bishul after Bishul does not apply to it. This is like a Tosefta (15:14; one may bake what was baked and cook what was cooked).

Mishnah Berurah (113): The Shulchan Aruch forbids when it is not fully cooked even when the pot is off on the fire. When it is fully cooked, the Beis Yosef permits using a ladle even when it is on the fire; Eliyahu Rabah is stringent.

Chazon Ish (15 DH Siman): If one wants to leave it on a fire that is not covered, since he may not return it if he takes it off, one may be lenient to take with a ladle.

Kaf ha'Chayim (175): If one needs to keep the food on the fire, he may remove it with a spatula, even if this will stir the contents.

Rema: L'Chatchilah, one should be careful about a pot in every case.

Taz (23): Mahariyo brings a proof from Shabbos 18b. This is not a proof. There it discusses wool, which is more stringent (for it needs much Bishul; or, one is liable for dying it). One may stir something that was fully cooked. This is why one may dice porridge in the pot with a wooden spoon (321:19), i.e. after it is fully cooked. The Rema is an excessive stringency. Therefore, I permit using a ladle to remove from a pot without intent to stir the pot. Even though this is forbidden if it is not fully cooked, one need not be careful about this if it is fully cooked. In 321:19, we permit l'Chatchilah! One who wants to be stringent will tilt the pot.

Magen Avraham (44): People are lenient about taking legumes out with a ladle, for there is no alternative.

Mishnah Berurah (117): The custom is not to be stringent like the Rema. One may even stir something fully cooked. It suffices to be stringent not to stir, but one need not be stringent about removing food with a ladle when it is not on the fire.

Kaf ha'Chayim (177): If it is easy to pour without a ladle, one should do so. If it is not Yad Soledes Bo (too hot to hold, the minimal temperature at which Bishul occurs), there is no concern.