My wife asked me the following Sheala:
If a pot of soup was taken off the stove on Shabbos and the lid (the underside of which was covered with steam water) was removed. Then after the steam water cooled down, one wants to replace the lid on the pot. The soup is still Yad Soledes Bo. So the water will drip down into the pot and will be cooked.
I thought that there might be basis to permit it based on the Rama 318:18 that says that water which was once Yad Soledes Bo and has cooled down may be reheated if it is not completely cold. Rav Moshe Feinstein ztl writes in a Teshuva (Igros Moshe 4:74) that the Rama really holds that Ein Bishul Achar Bishul but is Machmir (that Yesh Bishul Achar Bishul) for when it is fully cold. Based on this, the drops were presumably once Yad Soledes Bo. The drops falling into the soup is Psik Reisha delo Ichpas Lei. Also, there is the opinion of the Aruch HaShulchan and Eglei Tal that the view of the Rambam is that a Kli Rishon off the fire is only Mevashel Miderabanan. Although in general Pesik Reisha in a Derabanan is still Asur according to all but the Terumas HaDeshen (brought in Beis Yosef 253), in this case since it is only a Chumra according to Reb Moshe Feinstein maybe one could be maikel?
If any of my above reasoning is incorrect – I would be happy to hear from you.
You can be lenient to replace the lid.
The reasoning you suggest is correct, provided that the water on the lid is still hot enough to be considered “comfortably warm” for drinking. This is the level that Rav Moshe Feinstein defines as applicable for the ruling of the Rema you mention.
But beyond this reasoning, there is another reason for leniency, and this is that the condensed drops on the lid are “insignificant” for purposes of bishul. The Gemara writes concerning various Shabbos prohibitions (such as muktzeh) that something “insignificant” is not included in the prohibition, and several authorities have applied this idea to drops remaining in a cup after tea was drunk (for instance).
We can add this idea as another reason for leniency in the case of the question, so that it will be permitted to return it.