1. FOR THE DOUGH: Process flour, sugar, and yeast in food processor until combined, about 3 seconds. With processor running, slowly add water; process dough until 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 satiny, sticky ball that clears sides of work bowl, 30 to 60 seconds. 2. Transfer dough to lightly oiled counter and knead until smooth, about 1 minute. Shape dough into tight ball and place in greased bowl. Cover with plastic wrap and let rise at room temperature until almost doubled in size, 2 to 2 1/2 hours. 3. FOR THE SAUCE: Process all ingredients in clean, dry food processor until smooth, about 30 seconds; set sauce aside. (Sauce can be refrigerated for up to 2 days or frozen for up to 1 month.) 4. FOR THE TOPPING: Adjust oven rack to lowest position and heat oven to 500 degrees. Combine cheddar and mozzarella in bowl. Using pastry brush, grease bottom and sides of 2 dark-colored 9-inch round cake pans with 1 1/2 teaspoons oil each. 5. Transfer dough to lightly floured counter, divide in half, and shape into balls. Gently flatten 1 dough ball into 6-inch disk using your fingertips. Using rolling pin, roll disk into 10-inch round. Transfer dough to prepared pan and press into corners, forcing 1/4-inch lip of dough up sides of pan. Repeat with remaining dough ball. 6. Spread 1/3 cup sauce in thin layer over entire surface of 1 dough. Using pastry brush, brush sauce over lip of dough. Sprinkle 1 cup cheese mixture evenly over pizza, including lip. Repeat with remaining dough, 1/3 cup sauce, and remaining 1 cup cheese mixture. 7. Bake until crust is browned and cheese is bubbly and beginning to brown, about 12 minutes, switching and rotating pans halfway through baking. To remove pizzas from pans, run offset spatula along top edge of pizza crust. Once loosened, slide spatula underneath pizza and slide pizza onto wire rack. Let cool for 5 minutes. Slice and serve.