Gemara

38b - Question: If one roasted an egg [near a kettle heated on a fire], what is the law?

Answer (Rav Yosef): He is Chayav Chatas.

Support (Mar brei d'Ravina - Mishnah): Any food that that was soaked in hot water before Shabbos, one may soak it in hot water on Shabbos;

If a food was not soaked in hot water before Shabbos, one may pour hot water over it on Shabbos, except for old salted fish or Spanish tuna fish, for rinsing them is their final Melachah (preparation).

145b: The chicken of R. Aba is soaked and soaked again [for Refu'ah.]

Rishonim

Rambam (Hilchos Shabbos 22:8): If something was cooked or soaked in hot water before Shabbos, one may soak it in hot water on Shabbos. Something cold from the beginning that was never in hot water, one may rinse it in hot water on Shabbos, unless this is its final Melachah. One may not soak it in hot water on Shabbos.

Rosh (3:11): Presumably, as long as something did not reach k'Ma'achal Ben Drusai, anyone who cooks it further is Chayav.

Tosfos (39a DH Kol): If something came in hot water before Shabbos, i.e. it was fully cooked, e.g. the chicken of R. Aba, one may soak it in hot water on Shabbos, even in a Kli Rishon. If not, one may rinse it. The Rashba (this should say Rashbam, based on Tosfos 42b DH Aval, Tosfos ha'Rosh, the Beis Yosef and Rashash) holds that Iruy [from a Kli Rishon] is like a Kli Sheni. We understand why one may rinse something not cooked; but not soak it in a Kli Rishon. However, R. Tam holds that Iruy is like a Kli Rishon. If so, the Heter to rinse it must be from a Kli Sheni. In a Kli Sheni, one may even soak it! The Ri says that one may not soak it in a Kli Sheni. Since the water is hot, it looks like Bishul. Alternatively, one may soak it in a Kli Sheni. The Mishnah discusses rinsing to teach that even rinsing is final Melachah for old salted fish.

Poskim

Shulchan Aruch (OC 318:4): 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.

Magen Avraham (15): One may rinse from a Kli Sheni, but 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.

Kaf ha'Chayim (71): Machatzis ha'Shekel says to be stringent about onions, for perhaps they cook, and the Taz (8,14) is stringent. R. Zalman says that the custom is to be lenient to put them in a hot Kli Sheni.

Kaf ha'Chayim (70): Also the Bach permits only rinsing from a Kli Sheni. The Shulchan Aruch in Hilchos Pesach clearly holds like the stringent opinion [that Iruy Kli Rishon cooks Kedei Kelipah, i.e. an outer layer that can be peeled off].

Mishnah Berurah (34): One may not soak it in hot water that is Yad Soledes Bo, even if he will not leave it until it dissolves. We forbid even in a Kli Sheni, for this looks like Bishul. In Sa'if 9 we permit spices, for they are to give taste, so it does not look like Bishul.

Mishnah Berurah (35): One may rinse from a Kli Sheni. Iruy Kli Rishon cooks Kedei Kelipah. Likewise, one may pour from a Kli Sheni onto a cold wet food. This is like rinsing.

Kaf ha'Chayim (69): Rinsing a cold wet food is not Bishul. We distinguish between wet and dry foods only regarding soaking, but not regarding rinsing.

Shulchan Aruch (ibid.): However, old salted fish and Spanish tuna fish are forbidden, for rinsing them is their final Melachah.

Beis Yosef (DH Kulyas): Rashi explains that Spanish tuna is too salty to eat until it is rinsed in hot water. Rinsing it is its final Melachah. One is liable for this for Bishul. This connotes if it was cooked before Shabbos, one may soak it. He did not explain whether it was fully cooked, or if k'Ma'achal Ben Drusai suffices. Tosfos (39a) says that it must be fully cooked, like it says on 145b regarding the chicken of R. Aba. Also R. Yerucham and Hagahos Maimoniyos (9:1) say so. The Rambam connotes that it suffices if it was soaked in hot water before Shabbos, even if it was not cooked k'Ma'achal Ben Drusai. He says so to explain why the Mishnah says "came in hot water", and not "cooked". This is why Rashi (39a DH Kol) says "any salted food that came in hot water before Shabbos, one may soak it in hot water on Shabbos. This is not Tikun, for it was already partially fixed." I.e. "what was soaked in hot water" includes salted and unsalted food. For salted food, "came in hot water" refers to mere soaking. Regarding unsalted food, it refers to cooking. The Mishnah said "came in hot water", and not "cooked", due to salted food. The Rambam did not want to say so. Therefore, he wrote Stam "if something was cooked or soaked", without distinguishing salted from unsalted.

Taz (5): The Shulchan Aruch permits in cold water. One may wash herring in cold water, even if it is very salted. The Beis Yosef says so about tuna.

Magen Avraham (16): Something so salted that one cannot eat it is like something hard. One is liable for rinsing it.

Machatzis ha'Shekel: Rashi's Sefer ha'Pardes permits rinsing only what can be eaten without rinsing, and only in cold water. The Beis Yosef questioned this. The Magen Avraham holds that the Beis Yosef permits rinsing anything in cold water, and what can be eaten without rinsing, even in hot water. The Taz holds that the Beis Yosef did not rely on his reasoning to rule unlike Sefer ha'Pardes.

Kaf ha'Chayim (73): The Acharonim are stringent like the Taz to rinse only in cold water.

Mishnah Berurah (36): Several Acharonim say that one cannot eat these matters until they are rinsed in hot water. Therefore, rinsing them is considered Bishul, for it makes them a food. However, if one can eat something through rinsing in cold water, one may rinse it in hot water from a Kli Sheni. If so, one may rinse herring in hot water. The Taz forbids. Shulchan Atzei Shitim and Chayei Adam agree. This is proper.

Mishnah Berurah (37): One may even soak it in cold water. One may make edible something that was not edible (324:4).

Kaf ha'Chayim (75): R. Zalman permits rinsing in cold water even if this makes it edible. Chayei Adam forbids.

Rema: The same applies to anything hard that cannot be eaten without soaking it. One may not soak it on Shabbos, for this is the final Melachah.

Gra (DH v'Hu): We learn from the above, like it says on 38b. This shows that the same applies to any final Melachah.

Damesek Eliezer: Even though one who rinses is not liable for Bishul, since rinsing salted fish is the final Melachah, one is liable. Anything that can be eaten only through soaking, soaking is its final Melachah. We learn from roasting an egg. One is liable, since this is its final Melachah, even though it is not so much Bishul. However, this is only for soaking in hot water. One may soak in cold water, even if this is the final Melachah, just like one may roast an egg in the sun. However, perhaps this is because one could eat an egg raw. This requires investigation. See Chayei Adam and Nishmas Adam.

Mishnah Berurah (38): This refers to something that could be eaten raw, but it is so hard and dry that it is not proper to eat at all without soaking in hot water. Why did the Rema need to conclude that it is the final Melachah? It suffices that it was not cooked before Shabbos! I answer that he teaches that there is a Torah Isur to soak it, just like rinsing old salted fish.

Mishnah Berurah (39): The Poskim clearly say that Bishul applies to tea. One is Chayav Misah for this b'Mezid, or Chayav Chatas b'Shogeg. Surely, if one pours from a Kli Rishon onto tea leaves, there is concern for an Av Melachah. We hold that Iruy cooks Kedei Kelipah (Sa'if 10). All the more so, if one puts them on an oven until it is Yad Soledes Bo, he will be Chayav Misah. One may not even put the tea leaves in a Kli after pouring hot water into it, especially since some soft things get cooked through rinsing from a Kli Sheni. Perhaps there is a Torah Isur regarding the leaves. Therefore, one should do Iruy Kli Rishon on the leaves before Shabbos, so they will be somewhat cooked, i.e. Kedei Kelipah. He flips the leaves over while pouring on them. It is even better to properly cook them in a place where things cook. Afterwards, he empties the essence to another Kli. On Shabbos he may pour hot water over the dry leaves from a Kli Rishon. Later, he may return the essence to the cup with the leaves, for it is a Kli Sheni.

Sha'ar ha'Tziyun (63): The Pri Megadim questioned this. Perhaps Kedei Kelipah of the leaves was not fully cooked before Shabbos, and now it will cook more! Perhaps we permit due to a Sefek-Sefeka. Perhaps the Halachah follows the Rambam, who permits soaking on Shabbos what was soaked before Shabbos. Even if the Halachah does not follow him, perhaps it follows those who say that Bishul does not apply after a food reached k'Ma'achal Ben Drusai.

Mishnah Berurah (39): If one poured on the leaves from a Kli Rishon before Shabbos, one may pour n them on Shabbos, but he may not put them where they can get cooked, for they were cooked only Kedei Kelipah. One may rely on what I wrote. However, it is best to avoid pouring on the leaves on Shabbos. Rather, he pours hot water into the Kli in which he will drink, and then adds the cold essence to it. Likewise, one may put cold milk into this cup. If the essence is not cold, surely it is best to do so to fulfill all opinions.

Sha'ar ha'Tziyun (64,65): We are not concerned for coloring food (320:19,20). If tea essence is still warm, it is better to pour the [hot] water into the essence and not vice-versa, to fulfill the opinion that is concerned for coloring. See Ya'aros Devash (2, Drush 8).

Kaf ha'Chayim (65): The custom in Yerushalayim is to cook the leaves before Shabbos and leave liquid with them, and pour into the liquid on Shabbos, to avoid coloring.