Fried Neopolitan Pizzas

Stir together yeast and water in a measuring cup and let stand until foamy, about 5 minutes. (If mixture doesn't foam, discard and start over with new yeast.)
Stir together salt and 1 cup flour in a large bowl, then add yeast mixture and 1 tablespoon olive oil and stir until smooth. Stir in just enough remaining flour (1/4 to 1/2 cup) for dough to come away from side of bowl.
Turn out dough onto a lightly floured surface and knead, sprinkling with just enough flour to keep dough from sticking, until smooth, soft, and elastic, 8 to 10 minutes. Dust dough all over with flour and put in a bowl. Cover bowl with a kitchen towel or plastic wrap and let dough rise in a warm, draft-free place until doubled in bulk, 1 1/2 to 2 hours.
Heat oil (1/4 cup) in a 12-inch heavy skillet over moderately low heat until hot, then cook crushed garlic, stirring, until pale golden, about 1 minute. Add tomatoes and increase heat to moderately high, then cook, stirring occasionally, until sauce is thickened, 12 to 15 minutes. Stir in salt and pepper and transfer to a bowl. Discard crushed garlic if desired.
Put cheeses, basil, and minced garlic in separate bowls and keep, covered, at room temperature until ready to use.
Heat 1 inch oil in a deep 12- to 13-inch heavy skillet over moderate heat until it registers 350°F on thermometer.
While oil heats, divide dough into 8 portions and stretch each portion with floured hands, pressing on work surface, to form 4-inch rounds (don't worry if they are irregularly shaped).
Fry rounds 2 at a time, turning over once with tongs, until golden, 2 to 3 minutes per batch, and transfer to a paper-towel-lined platter to drain. (Return oil to 350°F between batches.)
Spread each fried dough round with some sauce and sprinkle with toppings. Pizza should be eaten immediately, folded in half.