원하다 means to desire. Vst고 싶다 is to want to V. (하다 + 고 싶다 = 하고 싶다, to want to do)
Considering 원하다 means more of "desire" than "want to do/eat/etc.", the example sentence is just hilarious.
BTW, "누가 불고기를 먹고 싶어요?" is the correct sentence. Better: "누가 불고기를 잡수시고 싶으세요?" would be the most natural.