Lightly coat a large bowl with 1 teaspoons of the oil. Put the dough in the bowl, cover loosley, and let sit at room temperature until supple and relaxed. 45 minutes to an hour.
Position racks in the top and bottom thirds of the oven and heat the oven to 475 degrees. Meanwhile put the tomateos, oregano, garlic, 2 teaspoons of the oil, and 1/2 tsp of salt & pepper in a food processor; pulse to make a chunky sauce.
Lightly oil 2 baking sheets with the remaining 2 teaspoons oil Divide the dough into quarters. Divide each quarter into 3 equal parts; you'll have twelve 1 ounce pieces of dough. Shape each piece of dough into a 3 inch round and put on the baking sheets.
Spread about 1 scant tablespoon 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 & prociutto. Sprinkle the mozzarella and Parmigiano on top.
Bake until the cheese is bubbling and the crust in browned, 12 to 14 minutes, rotating and swapping the pans positions about halfway through for even baking. Sprinkle with crushed pepper flakes if desired and serve.