Sprinkle yeast and sugar over water in large bowl of heavy-duty mixer; stir to dissolve. Add salt and mix to blend. Add 2 1/2 cups flour and oil and mix until dough forms ball and cleans sides of bowl, adding more flour 1 tablespoon at a time if necessary, about 5 minutes. Continue mixing until dough is smooth and elastic, about 10 minutes.
Cover dough and let stand in warm draft-free area 1 hour.
Mix all remaining ingredients except cheese in processor until smooth, stopping to scrape down sides of bowl, about 1 minute. (Sauce can be prepared 5 days ahead, covered and refrigerated.)
Turn dough out onto lightly floured surface. Divide in half. Pat each half into a round. Let stand 10 minutes.
Preheat oven to 475 degrees F.
Roll each round out on lightly floured surface into circle 1/8-inch thick. Transfer to cake rack or any flat perforated pan. Spread with sauce. Sprinkle with cheese. Bake on rack until edges of pizzas are lightly browned, about 10 minutes. Serve pizzas immediately.