I have doubts about the English version here. My answer was "boil for me the potatoes please" (since it is the potatoes and not "me" that are boiled), but I am not sure it is correct either.
it's a fine line between translating the word right vs translating the sentence right. in english I guess you'd say "make me a sandwich or make me potatoes". in russia you'd say "cook me x" if you need something cooked. or "fry me X" if it should be fried. but the meaning is the same "make me some food!!"
I agree. But Russian is an inflected language, so the difference between "свари мне картошку"/"свари меня картошке" is clear by the case endings. In English it is conveyed by word order and prepositions: e.g., a typical Russian mistake is saying "explain me X" instead of "explain X to me". But if, as you say, boil follows the pattern of "cook me X/make me Y" - it answers my question.