Don't move the dough during the first minute of grilling or it will tear. The preheated oven keeps the first pizzas warm while you grill the remaining rounds. If you like pepperoni, arrange 2 ounces of pepperoni slices over each pizza after you've topped it with sauce. Makes four 9-inch pizzas.
For the sauce:
Combine all ingredients in medium bowl. Set aside.
For the dough:
Whisk water, oil, sugar, and yeast together in 4-cup liquid measuring cup. Let sit for 5 minutes.
Pulse flour, Parmesan, and salt in food processor until combined, about 3 pulses. With processor running, slowly pour in water mixture and process until dough pulls away from sides and forms shaggy ball, about 1 minute. (If dough seems too sticky, add up to 2 tablespoons extra flour.) Turn dough onto floured counter and knead 3 or 4 times until cohesive.
Adjust oven rack to middle position and heat oven to 200 degrees. Line rimmed baking sheet with parchment paper and dust with flour. Divide dough into 4 equal pieces. Press 1 piece into small circle. Using rolling pin, roll and stretch dough to form 9-inch circle. Transfer to prepared sheet and dust with flour. Repeat with remaining dough, stacking each round on floured parchment.
Grilled pizza gets its charred, smoky, wonderful crust from a very hot fire. Unfortunately, if you're making two pizzas at once, the timing can be tricky: It's tough to flip and top them before they burn. Our two-level heat method takes the stress out of the equation.
If using a charcoal grill: Open bottom vent completely. Light large chimney starter filled with charcoal briquettes (6 quarts). When top coals are partially covered with ash, pour evenly over half of grill. Set cooking grate in place, cover, and open lid vent completely. Heat grill until hot, about 5 minutes. See steps 1 to 3 below.
If using a gas grill: Turn all burners to high, cover, and heat grill until hot, about 15 minutes. Leave primary burner on high and turn other burner(s) to medium-low.
1. Clean and oil cooking grate. Brush tops of 2 dough rounds lightly with oil. Peel rounds off parchment and place, oiled side down, on cooler side of grill. Grill, covered, until undersides are spotty brown and tops are covered with bubbles, poking large bubbles with tongs, 3 to 5 minutes.
2. Brush each round lightly with extra oil and flip.
3. Top each round with one-quarter of sauce and one-quarter of mozzarella. Grill, covered, until undersides are spotty brown and cheese is melted, 3 to 5 minutes. Slide pizzas to hotter part of grill to crisp, about 1 minute.
Place pizzas on wire rack set in second rimmed baking sheet and transfer to oven. Repeat with remaining rounds. Serve.