Bishul 21 - Ain Bishul Acher Bishul 1 (Klal 20 Siman 6) Hilchos Shabbos - S0518

We are beginning siman 6. The Chayei Adam moves from discussing sources of heat to discussing the items which are being cooked. 

We have three realms to discuss:
Something which is inedible raw and needs to be cooked;
Something which has already been cooked and needs to be reheated or cooked further, known as bishul achar bishul;
Something which is edible raw, but it changes by being cooked.

The Chayei Adam writes that after having discussed the heat sources, he will discuss the types of foods to which bishul applies. He begins with the second realm. There are two iterations of bishul achar bishul. One is where the food has already been cooked to the point at which it is edible, but cooking it again makes it more edible. The other is something which is fully cooked, and cooking it again is simply for the purpose of reheating it. 

The Chayei Adam begins with discussing the second case. The Chayei Adam writes that bishul does not apply to a solid food which is fully cooked on Erev Shabbos, because we have a concept that ain bishul achar bishul. This applies even if the item has completely cooled down. Since there is no further bishul possible, it is even muttar to place it in hot water in a kli rishon. Additionally, there is no issur derabanan around reheating it, since it does not even give the appearance of cooking, because it was fully cooked beforehand. Thus, one could add it to soup which is still a kli rishon. However, one cannot put it into a pot on the fire  or directly on the fire.

However, if this solid food was not fully cooked before Shabbos, it cannot even be placed into liquid in a kli sheini. Thus, for example, one cannot add it to the soup even after the soup was placed in a kli sheini. We mentioned the concept of kalei habishul, which would add to our concern because of the possibility of actual bishul. However, the concern over here is an issur derabanan that even without actual bishul, it will appear like bishul, which is known as mechzi k’mevashel. This concept is based on a Tosfos. Although other rishonim disagree, and some poskim do not accept this idea, the Magen Avraham, Chayei Adam and Mishnah Berurah accept this concept. Mechzi k’mevashel does not apply to something fully cooked, which is why in the previous case, where the item is fully cooked, there is no concern for recooking the food.

Summary
A solid food which is fully cooked can be reheated in liquid, even in a kli rishon and even if it fully cooled down, because ain bishul achar bishul.
A solid food which is not fully cooked cannot be reheated in liquid, even in a kli sheini, due to an issur derabanan that it appears like bishul (mechzi k’mevashel).