To make the meat crust: Preheat the oven to 450◦F (225◦C).
In a large bowl, mix the beef, onion, garlic, egg, cheese, salt, and pepper. Pat evenly in a 16-inch (40-cm) rimmed pizza pan.
Bake for 10 minutes. Remove from the oven and carefully pour out the juices that have accumulated in the pan. The meat will have shrunk to about 12 inches (30 cm), and this is fine.
To make the mushroom and pepper topping: Heat the oil in a large frying pan over high heat. Add the mushrooms, season with salt and black pepper, and fry for 5 to 10 minutes or until the mushrooms are browned and the liquid has evaporated. Scoop the mushrooms out of the pan into a sieve to drain. Add the red pepper to the pan. Fry until the peppers are slightly limp. Add to the mushrooms.
Preheat the broiler. Spread the tomato sauce evenly over the meat crust and scatter the mushrooms and peppers over it. Cover evenly with the cheese and sprinkle with the oregano (if using).
Broil until bubbly and lightly browned. Cool on a rack for a few minutes before slicing and serving.