Let dough stand covered at room temperature for 30 minutes. Heat the oven to 450°F with a rack in the lower third and brush an 18-by-13-inch rimmed metal sheet pan with 1 1/2 tablespoons olive oil.
Dust hands lightly with the measured flour and stretch dough across the pan. Let it rest for 5 minutes if it springs back, then press gently toward the corners to make an even thin layer.
Prick the center several times with a fork and bake the bare crust for 7 minutes. Toss thin mushroom slices with remaining oil while it bakes.
Spread pizza sauce over the par-baked crust, leaving a narrow border. Add mozzarella, pepperoni, mushrooms and oregano in a light even layer.
Bake for 12 to 16 minutes until the bottom is browned and the cheese bubbles. Scatter Parmesan over the hot pizza, rest for 3 minutes and cut into 12 rectangles, serving 2 per portion.