A crisp-yet-fluffy crust topped with a no-cook tomato sauce and melty, gooey cheese . . . welcome to pizza night.
Spray rimmed baking sheet with vegetable oil spray. Use pastry brush to evenly coat baking sheet with 2 tablespoons oil.
Place pizza dough on greased baking sheet and turn to coat with oil on both sides. Use your hands to pat and stretch dough into 10-by-6-inch rectangle. Cover with plastic wrap and let rise in warm place until bubbly and doubled in size, 1 to 1½ hours.
Use your hands to gently pat and stretch dough into corners of baking sheet. Cover loosely with plastic wrap and let rise in warm place until puffy, about 45 minutes.
After dough has risen for 1 to 1½ hours, use your hands to gently pat and stretch dough out to corners of baking sheet. (If dough snaps back when you press it to corners of baking sheet, cover it with plastic wrap, let it rest for 10 minutes, and try again.)
While dough rises, adjust oven rack to lowest position and heat oven to 450 degrees. Set colander in sink. Pour tomatoes into colander. Shake colander and drain well.
Transfer drained tomatoes to food processor. Add remaining 1 tablespoon oil, garlic, vinegar, oregano, salt, and pepper. Lock lid into place. Process mixture until smooth, about 30 seconds. Stop food processor. Carefully remove food processor blade (ask an adult for help). Transfer sauce to medium bowl.
When dough is ready, spoon sauce over dough and use back of spoon to spread into even layer, leaving ½-inch border around edges. Evenly sprinkle mozzarella and Parmesan cheeses over top of sauce.
Place baking sheet in oven and bake until cheeses are well browned and bubbling, 15 to 20 minutes.