Lightly coat a large bowl with 1 tsp. of the oil. Put the dough in the bowl, cover loosely, and let sit at room temperature until supple and relaxed, 45 minutes to 1 hour.
Position racks in the top and bottom thirds of the oven and heat the oven to 475°F. Meanwhile, put the tomatoes, oregano, garlic, 2 tsp. of the oil, and 1/2 tsp. each salt and pepper in a food processor; pulse to make a chunky sauce.
Lightly oil 2 baking sheets with the remaining 2 tsp. oil. Divide the dough into quarters. Divide each quarter into 3 equal parts; you’ll have twelve 1-oz. pieces of dough. (Alternatively, use a scale to divide the dough.) Shape each piece of dough into a 3-inch round and put on the baking sheets.
Spread about 1 scant Tbs. of the sauce over each dough round, leaving a little border around the edge for the crust. Top each with a mound of arugula, then some strips of pepper and prosciutto. Sprinkle the mozzarella and Parmigiano on top.
Bake until the cheese is bubbling and the crust is browned, 12 to 14 minutes, rotating and swapping the pans’ positions about halfway through for even baking. Sprinkle with crushed red pepper flakes (if using) and serve.