I know a bit about my therapist, the rest I have gathered from my (extensive) googling of her.
It's not that she's opposed to sharing but she did that more in the beginning, not so much now.
Like you, I want to know everything about her: what she likes to eat, watch, read, listen to, etc.
Would knowing more about her stop your curiosity a bit or would it feed it?
I know for me, nothing she may tell me will stop how curious I am about her: the more I know the more I want to know. It's not super healthy.
Also sometimes knowing too much might hurt you (as you say, you already envy her).
Personally, I wish I could unknow all the stuff about her kids that I found out.

Think of some questions you could ask her but my advice would be to not want to know too much.