My idea is similar to Laura's. I think it could work to create something similar to moderator elections:
A new question is created on Meta, where the "answers" are questions submitted by users. Other users upvote the questions they want to see answered. After a period of a week or so, the Q&A is locked.
However, instead of a livestream with a live chat, it might be better to create a new Meta post with the answers to the upvoted questions.
That way, there's a bit of a buffer, where overly-harsh or offensive questions are pushed to the bottom, and the ones the community most wants answered are the ones that actually get a response. Livestreaming it would open staff up to a lot of angry comments in realtime.
If there needs to be a follow-up round, there can be another post with the same process. I just personally don't think a live interaction would go very well, even if it could potentially resolve the situation more quickly. Also, even if the answers a pre-prepared, it's possible for the people reading the answers out live to misspeak or interject their own opinions without a review process.
Of course, the answers should definitely start with the questions that have already been asked and highly upvoted/discussed already. Compiling future questions into one Q&A, though, means Meta is a little less cluttered (not really the right word), and it's easy to browse through questions you might want to see answered.