Remove pizza dough from the refrigerator at least 20–30 minutes before grilling and let it rest on a lightly floured surface, loosely covered. This relaxes the gluten and makes the dough easy to stretch without tearing.
Preheat your grill to medium-high heat (425°F–500°F / 220°C–260°C) for at least 10–15 minutes with the lid closed. Scrub the grates clean with a grill brush, then wipe with an oil-soaked paper towel using tongs.
Stir together 2 tablespoons of olive oil and the minced garlic in a small bowl. Pat the mozzarella dry with paper towels and set out all toppings near the grill so you can work quickly once the dough is on.
Divide the dough in half and stretch each portion on a lightly floured surface into a rough 10–12 inch round about ¼ inch thick. Transfer to parchment paper dusted with cornmeal and brush the top with olive oil.
Flip the parchment paper over the grates so the oiled side lands directly on the grill, then peel off the parchment. Close the lid and grill for 2–3 minutes until the dough puffs, bubbles, and has visible grill marks underneath.
Flip the dough so the grilled side faces up, then immediately brush with garlic oil and spread a thin layer of pizza sauce, leaving a ½-inch border. Scatter on the mozzarella, Parmesan, red pepper flakes, salt, and pepper.
Close the grill lid and cook for 3–5 more minutes until the cheese is melted and bubbling and the crust bottom is golden brown. Slide the pizza onto a cutting board using tongs and a pizza peel.
Let the pizza rest 1–2 minutes, then scatter fresh basil over the top, drizzle with the remaining tablespoon of olive oil if desired, and slice. Repeat with the second dough portion, re-oiling the grates first.