Oh hmm. I hadn't even thought about mozz liquid acting as a sauce component. That's a helpful clue.

I'm not actually worried about evenness. My dough was ending up pretty close to as even as I cared it to be with stretching by hand; the really bad deformations happened in the process of moving it from breadboard to baking stone, and probably would have happened even if I'd used a stiffer dough and a rolling pin. Once I have better control of the peel, maybe I'll re-evaluate the dough-shaping stage.