At least 1 hour before baking, heat oven to 550°F (or 500°F if that's your oven's maximum temperature), with a baking steel or stone on the upper-middle rack.

In a food processor, process the tomatoes, oregano, garlic and pepper flakes until smooth, about 30 seconds. Transfer to a fine mesh strainer set over a medium bowl. Let stand without stirring until liquid no longer drains off, about 15 minutes. Discard the liquid. Add the strained tomatoes to the now-empty bowl and stir in the oil and salt. In a medium bowl, toss all 3 cheeses until combined.

Dust a baking peel, inverted baking sheet or rimless cookie sheet with semolina. Transfer the first dough round to the peel and, if needed, reshape into a 10-inch circle. Using the back of a spoon, spread half of the tomato mixture over the dough, leaving a ½-inch border at the edge. Sprinkle with half of the cheese mixture.

Slide the pizza onto the baking steel and bake until well browned and the cheese is bubbling and spotted brown, 7 to 10 minutes (9 to 12 minutes in a 500°F oven).

Using the peel, transfer the pizza to a wire rack. Let cool for a couple minutes. Meanwhile, repeat with the second dough round.