Preheat over 500°F with rack in upper third of oven.
Mix water, sugar, oil and yeast in a measuring cup. Allow to proof 5 minutes.
Pulse flour, Parmesan and salt in food processor. While running, drizzle in the water mixture until it forms a ball - about 1 minute.
Dump onto floured surface and knead a few times until it feels good. Divide in half.
Use a rolling pen to make 2 rounds about 11 inches in diameter - go for thinness more than roundness.
Transfer to oiled pan. Brush crust outside with olive oil.
Add toppings.
Bake on top rack until lightly golden, rotating halfway, about 10 minutes for cheese and 12 minutes for other.
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