In food processor fitted with metal blade, process flour, sugar and yeast until combined, about 2 seconds. With machine running, slowly add water through feed tube; process until dough is just combined and no dry flour remains, about 10 seconds. Let dough stand for 10 minutes.
Add oil and salt to dough and process until dough forms a satiny, sticky ball that clears sides of workbowl, 30-60 seconds. Remove dough and knead briefly on lightly oiled countertop until smooth, about 1 minutes. Shape dough into tight ball and place in large lightly oiled bowl. Cover tightly with plastic wrap and refrigerate for at least 24 hours and up to 3 days.
Process tomatoes, oil, vinegar, garlic, salt, oregano and pepper in food processor until smooth, about 30 seconds. Transfer to medium bowl and refrigerate until ready to use.
One hour before baking pizza, adjust rack in oven to second highest position, about 4-5 inches below broiler. Set pizza stone on rack and heat oven to 500 degrees. Remove dough from refrigerator and divide in half. Shape each half into smooth, tight ball. Place on lightly oiled baking sheet, spacing them at least 3" apart. Cover loosely with plastic wrap coated with non-stick spray; let stand for 1 hour.
Coat 1 ball of dough generously with flour and place on well floured countertop. Using fingers gently flatten into 8" disk, leaving 1" of outer edge slightly thicker than center. Using hands, gently stretch dough disk into 12" round, working along edges and giving disk quarter turns as you stretch. Transfer dough to a well floured peel and stretch into 13" round. Using back of spoon or ladle, spread 1/2 c tomato sauce in a thin layer over the surface of dough, leaving 1/4" border around edge. Sprinkle 1/4 c Parmesan evenly over sauce, followed by 1 c mozzarella. Slide pizza carefully onto stone and bake until crust is well browned and cheese is bubbly and beginning to brown, 10-12 minutes, rotating pizza halfway through. Remove pizza and place on wire rack for 5 minutes before slicing and serving. Repeat this with second ball of dough.