Gemara

15b - Mishnah - Beis Shamai: One must make two Tavshilim (for Eruv Tavshilin);

Beis Hillel say, one Tavshil suffices.

Beraisa - R. Eliezer: "Es Asher Tofu Afu v'Es Asher Tevashlu Bashelu" - one may bake only (relying) on an Eruv that was baked; one may cook only (relying) on an Eruv that was cooked. This verse is an Asmachta for Eruv Tavshilin.

16a - Abaye: The Mishnah says that one makes a cooked food from before Yom Tov. Bread is not valid.

Question: What is the reason?

Version #1 - Suggestion: We require an accompaniment (for bread, to show that it was prepared for Shabbos).

Rejection: Porridge is not an accompaniment, yet Abaye taught that it is valid!

R. Zeira derided Benei Bavel, for they eat bread with (porridge, which is made from grain just like) bread.

Answer: We require something uncommon. Bread is common, and porridge is not common.

Version #1 - Suggestion: We require something uncommon. Bread is common.

Rejection: Porridge is not common, yet Abaye taught that it is invalid!

Answer: We require an accompaniment. Bread is not an accompaniment, and neither is porridge.

17b - Beraisa - Chananyah - Beis Shamai: One may bake only if his Eruv included bread. One may cook only if his Eruv included a Tavshil. Hatmanah (wrapping food to keep it warm) is permitted only if he did Hatmanah before Yom Tov;

Beis Hillel say, one may rely on one Tavshil to do all of his needs.

(Mishnah - Beis Shamai): He must make two Tavshilim.

Our Mishnah is unlike the following Tana:

Beraisa - R. Shimon ben Elazar: Beis Shamai and Beis Hillel agree that two Tavshilim are needed. They argue about whether fish with an egg on top is considered like one Tavshil or two.

Rava: The Halachah follows the Tana of our Mishnah, according to Beis Hillel.

21b - Mishnah: R. Gamliel was stringent not to do Hatmanah on Yom Tov for Shabbos, like Beis Shamai.

Question: If he made Eruv Tavshilin, Beis Shamai should permit even Hatmanah. If he did not make Eruv Tavshilin, even Beis Hillel should forbid!

22a - Answer (Abaye): This is like Beis Shamai according to Chananyah. The Heterim to cook, bake and do Hatmanah are only for one who made bread, a Tavshil, and did Hatmanah, respectively. The case is, he made Eruv Tavshilin with bread and a Tavshil, but he did not do Hatmanah from before Yom Tov.

Rishonim

Rif and Rosh (2:5): We require an accompaniment. Therefore, bread and porridge are not valid.

Rif (10a): The Halachah follows the Tana of our Mishnah, according to Beis Hillel.

Rambam (Hilchos Yom Tov 6:3): One may not use bread or grits or similar things for Eruv Tavshilin.

Magid Mishnah: The Rif, Rambam, some Ge'onim, Ramban and Rashba do not require bread with the Tavshil. People are stringent to require bread to avoid the Safek

Hagahos Maimoniyos (2): The Yerushalmi requires a k'Beitzah of bread and a k'Zayis of a Tavshil.

Rosh (2:11): R. Tam says that Beis Shamai require two Tavshilim even just to cook; Beis Hillel say that one suffices. Our Mishnah discusses only cooking. Beis Hillel agree that one must prepare bread to permit baking, like R. Elazar (15b). Abaye said that bread is not valid, i.e. to permit cooking. Bread is needed to permit baking. Everything is included in 'Bishul' (cooking) - "U'Voshlu Es ha'Pesach" (Pesach is roasted). However, baking is not included. The decree against Bishul Nochri was very old; a Gemara thought that is mid'Oraisa! The decree against Nochri bread was among the 18 decrees and very well known. This shows that they were separate decrees. Chachamim require preparation only for the primary needs of a meal, but not for Hadlakah or Hotza'ah. Chananyah taught that Beis Hillel allow all needs of cooking relying on one Tavshil; this does not include baking.

Rebuttal (Ri, brought in Rosh, ibid.): Perhaps baking is included in cooking. Nevertheless, at first they did not decree against Nochri bread because most of the Tzibur could not fulfill it; the decree against Bishul Nochri was not so difficult. Not all agree with the Drashah "Es Asher Tofu..." A Tosefta teaches that one makes a cooked food from before Yom Tov and relies on it to cook, bake and do Hatmanah. In the Yerushalmi, R. Eliezer expounds from "Es Asher Tofu..." that one must bake and cook. R. Yehoshua learns from it that one may bake or cook. The Halachah does not follow R. Eliezer, for he is Shemuti (from Beis Shamai). Our Gemara did not bring R. Eliezer's opinion because the Halachah follows him, rather, because he expounds the source for Eruv Tavshilin.

Ran (10a DH Amar): R. Tam says that Beis Shamai and Beis Hillel argue about whether or not two Tavshilim are needed to roast and cook. Beis Shamai require two. Even though both are cooked, it is a reminder of two kinds of cooking. Beis Hillel say that one Tavshil permits roasting and cooking. All agree that bread is needed to permit baking, for it is not included in cooking. Our Gemara did not need to bring R. Yehoshua's opinion, because the Mishnah does not mention bread.

Rosh (ibid.): BaHaG and R. Shimshon also hold like R. Tam. The custom is to be stringent like this opinion.

Note: Tosfos (Sof 17b) says that the Ri himself was stringent like R. Tam.

Poskim

Shulchan Aruch (OC 527:2): The Eruv is made with bread and a Tavshil.

Source (Gra DH v'Im): R. Gamliel was stringent not to do Hatmanah on Yom Tov for Shabbos, like Beis Shamai (21b). Abaye establishes this according to Chananyah, who says that Beis Shamai say that the Heterim to cook, bake and do Hatmanah are only for one who made bread, a Tavshil, and did Hatmanah, respectively. The case is, the person baked bread and cooked a Tavshil but did not do Hatmanah from before Yom Tov. If Beis Hillel held that a Tavshil permits everything, the Mishnah could have (discussed one who only cooked a Tavshil and) said that R. Gamliel was stringent not to bake! Rather, we must say that also Beis Hillel require bread to permit baking. They argue only about Hatmanah.

Note: The Ri holds that Beis Hillel hold that a Tavshil permits even baking. Perhaps the Mishnah says that R. Gamliel was stringent only regarding Hatmanah because he always prepared bread and a Tavshil for Eruv Tavshilin, but he did not do Hatmanah from before Yom Tov because the food would not stay warm until Shabbos.

Shulchan Aruch (ibid.): If one made only a Tavshil it is permitted.

Mishnah Berurah (6): All agree that if one does not need to bake on Yom Tov for Shabbos, l'Chatchilah a Tavshil suffices.

Mishnah Berurah (7): If one prepared only bread, surely it does not permit to cook. The consensus of Acharonim is that it does not even permit baking, because Eruv Tavshilin was instituted primarily to be a Tavshil.