Make the Sauce: In a medium bowl, stir together the tomatoes, garlic, salt, sugar, pepper, and oil. (Alternatively, for a completely smooth sauce, you can blend the ingredients in a small food processor or blender.)
Preheat the oven to 500°F and set an oven rack in the bottom position. Dust a 13x18-inch baking sheet with half of the cornmeal.
On a lightly floured surface, using your hands, stretch and press one ball of dough into an 12-inch round. Transfer the dough to the cornmeal-dusted baking sheet, and gently stretch it out again so that it maintains its shape.
Spread about ¼ cup + 2 tablespoons of the sauce over the dough, leaving a ½-inch border around the edges. Slide the baking sheet into the oven and bake for 7 minutes, until the crust is partially cooked. Remove from the oven, and scatter half of the mozzarella cubes over the sauce, followed by half of the Parmigiano-Reggiano. Slide the pan back into the oven and cook until the crust is golden brown and the cheese is melted and bubbling, about 4 minutes more. Remove the pizza from the oven and transfer to a cutting board. Sprinkle with the basil and slice. Repeat with the remaining pizza dough and ingredients. (If you'd like to bake the two pizzas at the same time, you can fire up two ovens. Or, if you don't mind the crust a little thicker, you can form the two dough balls into 8x12-inch rectangles and place them side by side on the baking sheet.)
Note: This recipe makes enough sauce for 4 (12-inch) pizzas. Refrigerate extra sauce for up to 1 week, or freeze for up to 3 months.