Heat oven to 450° F with a rack in the lowest position. Coat a large rimmed baking sheet with 2 tablespoons of the oil. Stretch the dough into a large oval and place on the prepared sheet. Bake until the crust is firm but still pale, 8 to 10 minutes.
Remove the crust from oven and top with half the mozzarella, then the tomatoes, mushrooms, bell pepper, onion, pepperoni, the remaining half of the mozzarella, and the Parmesan. Bake until the crust is golden brown, 10 to 12 minutes more.
In a large bowl, toss the arugula with the vinegar, the remaining 2 tablespoons of oil, and ¼ teaspoon each salt and black pepper. Serve with the pizza.
Quick, easy, fresh, and healthy. No greasy pizza feeling. Used jarred roasted red peppers instead of fresh bell red pepper.