In an effort to avoid the problem of bishul akum, a Jew should be a part of the cooking process. In order to be using "aish yisrael" a Jew must turn on a flame, add something to the fire, or have created the fire from which the cooking flame is taken (all, for Ashkenzaim, as per this website's write up).
Does the Jew who lights the flame initially have to satisfy any standard of being a Jew? Could a child who is not yet bar mitzvah age qualify? What about a child who cannot formulate the intent* (he flails his arms and it hits a button which ignites, or even, a non-Jew moves a Jewish baby's hand)? Could it be someone who is a shoteh?
*the Shulchan Aruch (Y"D 113:7) talks about חתוי בלא כונה but I'm not sure if this means someone who COULD have intent, or what kind of intent he doesn't have (to turn on the flame, or to move his own hand etc) and it seems to be argued upon by commentators anyway.