1. Preheat oven to 500 degrees, with rack in lower third lined with a pizza stone or baking sheet. Drizzle a 13-by-18-inch rimmed baking sheet with 3 tablespoons oil. Turn dough out onto center of sheet (do not punch down). Starting in center and using both hands, press down and stretch dough outward with your fingertips, working to edges of pan in an even thickness. Loosely cover with plastic wrap or a clean kitchen towel and let rise again until doubled in volume, 30 to 45 minutes. (Dough should come about three-quarters up sides of sheet.)

2. Lightly drizzle dough with oil; season with salt and pepper. Bake on stone until puffed and beginning to set on top, 5 to 7 minutes. Meanwhile, if using mushrooms, toss with remaining 2 tablespoons oil. Remove dough from oven.

3. Spread 1 1/4 cups sauce evenly over dough. Sprinkle with mozzarella, then dollop with small spoonfuls of remaining 3/4 cup sauce. Top with mushrooms, pepperoni, or both. Sprinkle with oregano. Return to oven and continue baking until crust is puffed and golden brown in places and cheese is melted and bubbly, 18 to 20 minutes more (if browning too quickly, loosely tent with foil). Let cool in pan 5 minutes, then cut into squares and serve.