Stir in tomatoes and sugar, increase heat to high, and bring to a simmer. Lower heat to medium-low and simmer until reduced to about 2½ cups, 25 to 30 minutes. Off the heat, stir in the basil and olive oil, then season with salt and pepper.
Laminate the Dough: Turn the dough out onto dry work surface and roll into a 15×12-inch rectangle. Using an offset spatula, spread the softened butter over the surface of the dough, leaving a ½-inch border along the edges. Starting at the short end, roll the dough into a tight cylinder. With seam side down, flatten the cylinder into an 18×4-inch rectangle.
Cut the rectangle in half crosswise. Working with one half, fold into thirds like a business letter; pinch seams together to form a ball. Repeat with the remaining half. Return balls to the oiled bowl, cover tightly with plastic wrap and let rise in refrigerator until nearly doubled in volume, 40 to 50 minutes. Meanwhile, adjust oven rack to lower position and preheat oven to 425 degrees F.

Bake the Pizzas: Coat two 9-inch round cake pans with 2 tablespoons of olive oil each.
Transfer 1 dough ball to dry work surface and roll out into a 13-inch circle. Transfer dough to the pan by rolling the dough loosely around a rolling pin and unrolling into the pan. Lightly press dough into the pan, working into corners and 1-inch upsides. If dough resists stretching, let it relax for 5 minutes before trying again. Repeat with remaining dough ball.
For each pizza, sprinkle 2 cups mozzarella evenly over the surface of the dough. (If you’re using any meat or veggie toppings, add them now, on top of the cheese.) Spread 1¼ cups tomato sauce over the cheese (or toppings) and sprinkle 2 tablespoons Parmesan over the sauce. Bake until crust is golden brown, 20 to 30 minutes. Remove pizza from the oven and let rest 10 minutes before slicing and serving.