Is there a query I can run against a backup of our database to restore one thread and the posts associated with it?q
Yes I have backups (first part of the sentence in my post).If you permanently deleted it, it's gone forever . It's not easy to use data recovery tools on DBs either, sadly.
Have you got an older backup? If so, you could recover it to a temp DB and then extract that post/thread and reinsert it.