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

Place a colander in a bowl. Add the tomatoes and sprinkle with ½ teaspoon salt. Toss, then let stand, tossing occasionally to encourage juices to drain off, until ready to use.

Generously dust the counter with flour and set one portion of dough on top. With your hands, slightly flatten the dough, then with your fingertips gently press it into an 8-inch round of relatively even thickness. Flip the dough over and press into a 10-inch round, working from the center outward and leaving a ½-inch border around the perimeter; occasionally flip the dough to ensure the bottom is not sticking to the counter.

Dust a baking peel with semolina, then transfer the round to the peel and, if needed, reshape into a 10- to 12-inch round; gently shake the peel to ensure the dough slides freely. Tear half of the mozzarella over the surface of the dough, leaving a ½-inch border around the perimeter. Using your hands, gently squeeze about half of the tomatoes to remove excess moisture, then scatter the tomatoes over the dough. Sprinkle with half of the Parmesan and drizzle with oil.

Slide the pizza onto the baking steel and bake until the crust is spottily browned and the cheese is melted, 8 to 10 minutes. Using the peel, transfer the pizza to a cutting board. Brush away any residual semolina on the peel.

Shape, top and bake the second portion of dough in the same way. While the second pizza is baking, sprinkle the first pizza with ground black pepper and half of the basil. Cut into slices and serve. When the second pizza is done, remove it from the oven and finish it in the same way. Cut into slices and serve.