Someone started to cook food on a fire that he lit. A goy working in the house came over thinking the food was fully cooked and turned off the fire. The goy then realized that it wasn’t fully cooked (not even machel ben dorsa) and right away turned back on the fire, is this a problem of bishul akum?
If the food was not yet cooked, and the fire was turned off, the initial “turning on” of the fire by the Jew does not help, and this will constitute a problem of bishul akum.
The problem of bishul akum can be solved (according to Ashkenaz custom) by a Jew “taking part” in the fire that cooks the food, such as by the Jew turning up the fire, or even adding something small to the fire.
However, in this case, the Jew has not contributed anything to the fire that ultimately cooked the food, and therefore it will be bishul akum.
For a (paid) housekeeper, there is room for leniency in bishul akum, as we have explained here.