There appears to be a difference of opinion between contemporary authorities regarding whether and how a Jew's triggering the cooking process using some indirect electronic means grants the resulting food "cooked by a Jew" (bishul Yisrael) status. Here are some pertinent opinions that I found online, which may not be representative of the full gamut of authoritative positions. (All parenthetical jargon-translations are mine.)
The Orthodox Union (OU) provides a way to deem food produced in a factory bishul Yisrael through a Jew turning the ovens on by remote control:
Rabbi Yehuda Shain has recently developed an ingenious system whereby the mashgiach (kosher supervisor) can monitor the production from an off-site location. By installing a special device, it is possible to turn the oven on and off through the use of a touch-tone phone.
R' Yisroel Belsky, who is one of the OU's chief authorities, was cited by R' Eli Gersten providing more detail:
Rav Belsky said that it is
acceptable to have the Mashgiach turn on the boiler
remotely, from his cell phone, by dialing in a special
code. ... Also if the Mashgiach sets a timer in the
evening to turn on the fire the next morning this is also
acceptable. However, Rav Belsky only allows timers
for a one time use but not if they turn on the oven
again and again, because then the ma’aseh Yisroel (action of the Jew)
is lost. Rav Belsky said that a timer for its first use is a
real ma’aseh (action), similar to aisho mi’shum chitzo ("his fire [directly] via his arrow"), and is
not only a גרמא (indirect action), but subsequent uses are only a גרמא.
However, according to R' Moshe Heinemann, the Star-K's chief authority, the use such devices does not grant bishul Yisrael status:
If the action that is done by the Yehudi (Jew)
causes an indirect lighting of the oven, that action would not
qualify for bishul Yisroel. Hence, dialing a number that in turn
trips a switch that in turn lights an oven would be considered a
“grama”, an indirect action that would not qualify for bishul
Yisroel.
It seems to me that according to R' Belsky and presumably the OU, a Jew's programming the robot initially to cook many dishes in the future would not cause those meals to be considered bishul Yisrael, since the dishes would be only indirect results of the programming. However, if a Jew activates the robot to make an individual dish, then the resulting dish could be considered a direct result of the Jew's activation, like the timer set the night before in R' Belsky's example, and therefore bishul Yisrael. According to R' Heinemann, who doesn't consider a remote-controlled activation that immediately lights an oven sufficiently direct to grant bishul Yisrael status, I think it's unlikely that any sort of programming or activation of this robot, short of manually turning on its stove, would allow its product to be considered bishul Yisrael.
Of course, the laws of bishul Yisrael are complicated (and include many exceptions that are applicable to many practical cases), technology is complicated, and the application of the former to the latter is very complicated. If an issue of this sort comes up practically for you, consult your rabbi.
