Answer

I don’t have time to discuss this and work it out at length right now (I’m on the road and don’t have books with me), so I’ll address it briefly and from memory. When I have time, it would definitely be interesting to clarify this law.

See Sabbath 4a regarding someone who sticks bread dough to the wall of an oven. There the discussion is the opposite: if someone stuck bread in the oven, may he remove it in order to save himself from the prohibition of baking on the Sabbath, which is punishable by stoning. The conclusion is that he may do so in order to save himself. From that passage it emerges that sticking bread into an oven when the baking is completed on the Sabbath is a prohibition punishable by stoning.
Accordingly, if you extend the fire so that its surface will crust over, then apparently you have only made the situation worse, for even without that you would have been liable, so if you added something more with your own hands, that certainly would not exempt you.

However, that is not so, because you are discussing a case where, when you stuck the bread in, it was not supposed to crust over, and the crusting was caused by an action done afterward. Here the initial placement involved no prohibition, and afterward it is only indirect causation. And that is what you wrote, that it depends on whether such placing on the Sabbath itself would incur liability—because, as we saw, there is no difference as to when one sticks it in, so long as the crusting happens on the Sabbath.
Still, on logical grounds it seems to me that the indirect causation here is with respect to the kindling, not the cooking, because cooking is always done indirectly—you place it on the fire or light a fire under the pot, and the fire does the cooking. Therefore, there is apparently room to say that if you kindle a fire indirectly, then you would be exempt for the kindling but liable for the cooking (since with cooking there is no exemption for indirect causation). Unless perhaps indirect causation of indirect causation is different.
But on second thought, this situation is like the case of “his arrows have ceased” in Bava Kamma, in the passage about “his fire is due to his arrows.” There we find that if a person kindled a fire in a way that was not supposed to cause damage, and afterward performed an action that enabled the fire to spread and cause damage (like shooting an arrow toward a shutter and afterward moving the shutter), then according to Reish Lakish, who holds that liability for one’s fire is due to one’s property, he would be liable, but according to the opinion that liability for one’s fire is due to one’s arrows, he is exempt. In your case, this is like one who kindles in his own domain and not in another person’s domain, and in that case he is exempt. And see Nimukei Yosef there, who discusses whether Sabbath candles are comparable to kindling a fire in the law of damages. True, that is regarding the kindling, but regarding the cooking there is the distinction I mentioned above, that in this area one is liable even for indirect causation.

It seems to me that this could be connected to the discussion of the later authorities about someone who threw a vessel from a rooftop and then broke it a moment before it hit the ground (or threw a baby and someone below received him on a sword). There too some engaged in dialectical analysis to exempt him, because the one who receives it is exempt since he broke an already-broken vessel, but the one who threw it is also exempt, since its breaking was done by the one who broke it.
And similarly in the case of setting one’s own dog on another person’s ox, where some have exempted him (the Chazon Ish and Penei Yehoshua), according to the view that one who sets another person’s dog on a different other person’s ox is exempt.

The Rashash, the Maggid Mishneh, and other later authorities discussed sowing and baking/cooking, because in these two labors the mode of performance is always indirect. A person only places it on the fire or lights the fire, but the fire is what cooks. And so too with sowing (assuming that the realization of the prohibition is the growth that comes afterward, which is not our topic here). In such a case, regarding sticking it into the oven in advance, they discussed the question whether if one places it on the Sabbath and it gets cooked after the Sabbath, he is liable or exempt. If the day is entering, we saw that he is liable, but if the day is ending there are arguments both ways. The dispute concerns how to define the labor of cooking: is the placement on the fire the labor, with the completion of the baking being a condition (in which case even if it was completed after the Sabbath he is liable), or is the completion the labor and the placement only a condition (in which case it must be completed on the Sabbath in order to make him liable)?
According to the side that says the completion is the labor and the placement is a condition, it seems that here he would be liable, because the completion took place, and he fulfilled the condition through indirect causation, and in that there is no exemption. But according to the side that says the placement is the labor and the completion is a condition, there is reason to exempt him, because he performed the actual labor itself through indirect causation. According to that, there is really no true difference between kindling and cooking, because when he lights the fire he also violates cooking and not only kindling. True, one can also violate cooking by placing food on a fire that is already lit.

Now, of course, all this has to be weighed and summarized and brought to a conclusion (and checked to see whether I remembered correctly). Not for now.
 
Just one additional note that I suddenly noticed.
It seems from what you wrote that if you had not seen that an extension was needed, and the initial placement had been enough, then it is obvious to you that it would be permitted, and the whole question arises only because you saw that an extension was needed, and that is a reason to prohibit it. But notice that it is exactly the reverse. Without the need for an extension, it is obvious that it is prohibited. Sticking it in before the Sabbath when the cooking is done on the Sabbath is certainly prohibited by Torah law (a prohibition punishable by stoning, as in the passage in Sabbath there). On the contrary, the discussion begins only because when you placed it there, it was not supposed to bake, and about that I wrote all the Torah-level chatter above.
 
——————————————————————————————
Questioner:
Thank you for the detailed answer; it gave me food for thought and an important source sheet.
The issue of “his fire is due to his arrows” is definitely something I need to look into.
Regarding the last note, how do you know that he stuck the bread in before the Sabbath? הרי it says explicitly in the Talmud that one may leave a pot of raw food on the fire, because then there is no concern that he will stir it
——————————————————————————————
Rabbi:
I misspoke. You are of course correct.