Stand dough covered at room temperature 30 minutes. Heat oven to 450°F, rack in lower third. Brush 18-by-13-inch rimmed metal sheet with 1 1/2 tablespoons oil.
Lightly dust hands with measured flour; stretch dough across pan. Rest 5 minutes if springing back, then gently press toward corners into even thin layer.
Fork-prick center several times; bake bare crust 7 minutes. Meanwhile, toss mushrooms with remaining oil.
Spread sauce over par-baked crust, leaving narrow border. Add mozzarella, pepperoni, mushrooms, and oregano in light even layer.
Bake 12 to 16 minutes until bottom browned and cheese bubbling. Add Parmesan; rest 3 minutes. Cut 12 rectangles; serve 2 per portion.