How do enlightened people think? Most of our thoughts are caused by desire. So, what are the thoughts of a person who has no desire?

Another way of asking the same question: if we remove desire based thoughts, what will remain?

Edit: I'm asking this because I want to experience enlightenment. I know I am not enlightened, but If I am enlightened, how would my thoughts be? How would I feel it through awareness?
Enlightened people do not overgeneralize.

(This can be supported by both Pali Canon as well as Mahayana sources.)

Concepts like "enlightened", "person", "I", and "thinking" are all overgeneralizations, which enlightened people do not dwell in.

They may use generalizations as skilful means, to communicate the teaching, but they themselves are not bounded by them.