Does Bishul Akum require intent?

Question

In this answer, @AdamSimon suggested that:

Therefore, in our case (in which the power goes out on or before Shabbos and is subsequently restored during the course of Shabbos), provided that the majority of people for whom the power company is restoring power are gentiles and the majority of people who work for the power company on Shabbos are gentiles (like in most parts of America), there should be no issue with deriving benefit from food which was heated or cooked on Shabbos even if it was completely raw or cold prior to the return of power (provided that the heating element is properly covered so that a Jew would not be tempted to adjust it - pursuant to the Rabbinic restriction of grufa ve'katuma for shehiya and chazara).

Is the food that was cooked due to the actions of the non-Jew Bishul Akum? Or do we say that since the non-Jew had no intention to cook the food, it is not Bishul Akum?

The "Shain System" is used by Kosher Certification Companies to remotely ignite ovens, to ensure Pat Yisrael. Does the same thing apply here, in reverse?

Answer 1

This is what it says in Shulchan Aruch Yore Deah 113:5: If a non-Jew cooks but he did not intend to cook -it's permissible (e.g. he lit a fire in a field to get rid of the grass/hay and Chagavim were cooked -they are permitted).. But if his intention was to cook -it's prohibited (e.g. he lit a fire to bake bread and he didn't have any knowledge that there was meat in the oven -the meat is forbidden).

In this case the electric company had no intention of cooking so, I assume, it would not fall under the category of Bishul Akum.