Quarter dough and shape into 4 balls. Let rest at room temperature until no longer stiff, 20 to 30 minutes.
Meanwhile, heat a gas or charcoal grill to medium (350° to 400°). Brush dough balls liberally all over with olive oil, then set each on a parchment-lined baking sheet and gently stretch or roll into 10-in rounds (if it resists, let sit 10 minutes and try again).
Invert each round onto grill and peel off paper. Cook until browned and beginning to puff, 2 minutes. Return to baking sheet, grilled side up. Hold par-grilled dough for up to 2 hours at room temperature, or top and grill immediately.
For Lemon, Pecorino, and Red Onion pizza: Scatter pecorino over 2 par-grilled dough rounds, leaving a ½-in. border around rim. Arrange lemon slices over pecorino, followed by red onion rings. Sprinkle with rosemary leaves and sea salt, then finish with lashes of extra-virgin olive oil.
With a wide spatula, ease back onto medium-hot grill. Cook, covered, rotating often to cook evenly, until well browned and crisp underneath, 4 to 6 minutes.
For Biancoverde pizza: Scatter each par-grilled dough with 1/3 cup coarsely shredded Parmigiano-Reggiano cheese. Add sliced fresh mozzarella and dollops of ricotta. Grill pizzas as above, then top each with arugula. Sprinkle with pepper, salt, and extra-virgin olive oil.