Gemara

R. Chiya bar Aba went to Gavla. He saw Yisraelim eating lupines cooked by Nochrim. He did not say anything; he came to R. Yochanan.

R. Yochanan: Go announce that their lupines are Bishul Akum (food cooked by Nochrim, which is forbidden), since they are not Benei Torah.

Inference: If they were Benei Torah, their lupines would be permitted!

Question: Rav Shmuel bar R. Yitzchak taught that Bishul Akum does not apply to anything (normally) eaten raw. Lupines are not eaten raw!

Answer: R. Yochanan holds like the other version of Rav Shmuel's teaching:

(Rav Shmuel bar R. Yitzchak): Bishul Akum does not apply to anything that does not come on the table of kings to be eaten with bread.

Therefore, Benei Torah are permitted.

Avodah Zarah 38a - Question: What is the difference between the two versions of Rav Shmuel's teaching?

Answer: They differ about small fish, mushrooms, and porridge.

37b (Mishnah): Bishul Akum is forbidden.

Question: What is the source of this?

Answer #1 (R. Chiya bar Aba): "Ochel ba'Kesef Tashbireni v'Ochalti u'Mayim..." - just like we may drink water of Nochrim, which has not changed, also their food is permitted if it has not changed.

Objection: If so, it should be forbidden to eat Kelayos (wheat that was dried in an oven) of a Nochri!

Suggestion: Perhaps that is true!

Rejection (Beraisa): If a Nochri made Kelayos out of wheat, it is permitted.

Answer #2: Rather, just like we may drink their water, which is in its natural form, also their food is permitted if it is in its natural form.

Objection: If so, if a Nochri ground wheat, the flour should be forbidden!

Suggestion: Perhaps that is true!

Rejection (Beraisa): We permit flour and fine flour of a Nochri, and Kelayos that he made out of wheat.

Answer #3: Just like we may drink water of Nochrim, which has not changed through fire, also their food is permitted if it has not changed through fire.

Objection: The Torah does not say that it depends on whether or not it changed through fire!

Answer #4: Chachamim decreed against Bishul Akum. The verse is only an Asmachta.

Rishonim

The Rif (Avodah Zarah 14b) brings the two versions of Rav Shmuel's teaching and the difference between them.

Rambam (Hilchos Ma'achalos Asuros 17:14): If a Nochri cooked for us wine, milk, honey, quince or similar things that are eaten raw, it is permitted. Chachamim decreed only about things that are not eaten raw, such as meat, unsalted fish, eggs and Yerakos.

Rambam (15): This is if the food goes on kings' tables to be eaten with bread, such as meat, fish, eggs and similar things. Something that does not go on kings' tables to be eaten with bread, such as lupines, even if it is not eaten raw, is permitted. The same applies to all similar things. The decree was primarily to prevent intermarriage. It discourages inviting a Nochri to a meal. People normally do not invite friends to eat things that do not go on kings' tables to be eaten with bread.

Ran (Sof DH Rebbi): Alternatively, the decree against Bishul Akum was so Yisraelim will not regularly eat with Nochrim and come to eat forbidden food or drink. Since there is an Asmachta from a verse, it seems that the decree is patterned on the verse. Anything not changed by fire is permitted, such as parched grain, even if it is not eaten raw and goes on kings' tables to accompany bread. When the Gemara says that the decree is mid'Rabanan, it does not retract from what is said above that their food is permitted if it has not changed through fire. However, it is not a proper Drashah, only an Asmachta. This is why Kelayos are permitted, not because they are eaten raw. This is how Rashi learns, and it seems that the Rif agrees. However, the Rambam permits Kelayos because people do not invite friends to eat them.

Hagahos Maimoniyos (200): The second version agrees to the leniency of the first version. We find that Amora'im permitted cooked oil and water or Nochrim. Since R. Yochanan holds like the second version, the Halachah follows the leniencies of both versions. Avi ha'Ezri says that some are stringent like both versions.

Rosh (Avodah Zarah 2:28): R. Tam says that we hold like both versions of Rav Shmuel's teaching. Bishul Akum is forbidden only if the food goes on kings' tables and is not eaten raw. He infers from the fact that the Gemara illustrated the difference between them by listing things that are not eaten raw and do not go on kings' tables to be eaten with bread. The first version forbids them, and the second version permits them. It does not list things that are eaten raw and go on kings' tables and say that the first version permits them and the second version forbids them. Because we are lenient like both versions, it lists only things permitted by the second version. We hold that the second version always comes to add. Later, the Gemara says that honey is permitted because it is eaten raw. R. Yochanan permits anything k'Ma'achal ben Drusai (cooked a half or a third by Yisraelim), and all the more so something eaten raw. He also permits lupines because they do not go on kings' tables.

Note: R. Tam holds like Rashi, who says the differences listed (small fish, mushrooms and porridge) are not eaten raw and do not go on kings' tables. The Rashba says that porridge can be eaten raw but goes on kings' tables).

Ran (DH Kol): R. Chananel and the Rambam hold like R. Tam. The Ramban holds like the version that forbids anything not eaten raw, even if it does not go on kings' tables. He learns from cooked bitter or semi-bitter dates. Even though they do not go on kings' tables, Rav forbids them because they are not eaten raw (38b). This is not a proof. Perhaps even bitter dates go on kings' tables after they are sweetened through cooking! It seems that something is permitted if it is eaten raw, does not go on kings' tables, or was not changed through fire. However, I did not see Rishonim permit whatever was not changed through fire. The Rif brought both versions of Rav Shmuel's teaching and did not decide which we follow.

Poskim

Shulchan Aruch (YD 113:1): If a Nochri cooked something this is not eaten raw, if it goes on kings' tables to accompany bread or for Parperes (an appetizer or dessert), it is forbidden. This is even if he cooked it in a Yisrael's Keli in the Yisrael's house.

Shach (1): This is unlike the Ran, Rashi and Hagahos Ashri (Avodah Zarah 2:28), who says that we are concerned only for something that changed through fire. It seems that the Rif and Ra'avan agree with them. However, the other Poskim do not require that it changed through fire. The Rema agrees; he permits peas (Sa'if 2) only because they do not go on kings' tables.

Darchei Moshe (3, DH v'Chosav): Isur v'Heter says that something eaten raw in pressed circumstances is not considered something eaten raw. Even though one who eats innards is 'not a person' (Nedarim 74b) and they are not considered Chatichah ha'Re'uyah Lehiskaved, they are forbidden due to Bishul Akum. The same applies to mushrooms.

Gra (2,3): The Gemara explicitly discusses foods that go on kings' tables to accompany bread. The Gemara (Avodah Zarah 38a) said that a Nochri who roasted grasshoppers forbids them due to Bishul Akum; the Rambam explains that they are Parperes.