Bake pizza dough until it has just started to brown, and is almost done.
Place tomatoes on a baking sheet lined with parchment paper. Do the same with the mushrooms. Drizzle olive oil liberally over vegetables, 2-3 Tablespoons for each pan.
Drizzle with about half as much balsamic vinegar.
Season both pans well with salt and pepper and sprinkle each with 1 tsp sugar.
Roast at 400 about 20-30 minutes.
Remove mushrooms sooner if they are done. Drain tomatoes on paper towels
Top pizza crust with a layer of pesto sauce.
Cut mozzarella into pieces and arrange on dough.
Top with tomatoes and mushrooms.
Sprinkle with Parmesan cheese.
Bake at 400 until cheese is melted and pizzas are hot, about 10-15 minutes, watching carefully.