Can I use one job offer to speed up another offer (on my dream job)?
I recently applied for this position, and I think I'd be a great fit, but I have another offer in hand. I'd much rather be a part of University B's project, however. When might you know what candidates you are interested in?" 


That is almost perfect. Instead of asking when they will know, tell them when you need to know by. With two weeks notice it is not clear what they can do, but they probably could tell you if you are ridiculously strong (think an associate professor who is ready to be promoted to full professor applying for an assistant professor position), a reasonable candidate, or not in the running. You may want to talk to the university that made the offer and see if the deadline can be extended.

For an extremely qualified candidate with an application that stands out, and I find that a dubious statement, a university could move you to the top of the short list, interview you, and make an offer in under a week. That said, it is extremely unlikely that a search committee would be so impressed that it decides to move that fast.