In a medium bowl, gently toss together the sliced tomatoes, 2 tablespoons (30 ml / 1 fl oz) of snipped parsley, minced garlic, salt, and pepper.
Set this fresh tomato mixture aside to let the flavors meld.
Prepare your grill for direct, low heat. For a charcoal grill, arrange low coals. For a gas grill, preheat and then reduce the heat to low. Lightly grease the grill rack.
Divide the pizza dough in half. On a lightly floured surface, pat or roll each half into a 12-inch circle.
Brush the top of each dough circle with olive oil. To make grilling easier, stack the dough circles on a baking sheet, separated by parchment paper that has been sprinkled with cornmeal.
Carefully place one dough circle, oiled-side down, onto the grill rack.
Grill for about 6 minutes, uncovered, until the bottom is light brown and has nice grill marks. The top should look dry but still be soft.
Remove the crust from the grill and place it on your baking sheet, grilled-side up. Quickly top with half of the tomato mixture and half of all three cheeses.
Slide the pizza back onto the grill.
Cover and cook for another 4 to 6 minutes, or until the crust is crisp and all the cheese is gloriously melted and bubbly. Rotate the pizza as needed to ensure it cooks evenly.
Remove the finished pizza from the grill, sprinkle with fresh parsley or basil, and let it rest for a minute before slicing.
Repeat the process with the second dough circle and remaining toppings.
Let your refrigerated pizza dough sit at room temperature for about 20-30 minutes before rolling. This relaxes the gluten, making it much easier to stretch without snapping back.
Keep the grill on low to medium-low heat.
Grilling pizza dough directly over high heat can burn the crust before the top is cooked and the cheese is melted.
