Heat oven to 400° F. Coat a 9-inch oven-proof skillet or cake pan with 1 ½ tablespoons of the oil. Press the dough into the pan, covering the bottom and sides. Top with half the mozzarella, then the tomatoes, oregano, and remaining mozzarella.
In a small bowl, toss the onion, olives, mushrooms, and pepperoni with ½ tablespoon of the remaining oil; scatter over the pizza. Bake until the crust is golden brown, 30 to 35 minutes.
Meanwhile, in a large bowl, toss the arugula with the vinegar, the remaining tablespoon of oil, ½ teaspoon salt, and ¼ teaspoon pepper. Serve with the pizza.
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