In a medium bowl, gently combine the chopped tomatoes, snipped basil, olive oil, and minced garlic.
Set this fresh tomato mixture aside to let the flavors meld.
On a lightly floured surface, roll out your thawed bread dough into a 12-inch or 30cm circle. Lightly sprinkle both sides of the dough with cornmeal—this prevents sticking and adds a wonderful crunch.
Prepare your grill for medium heat. For a gas grill, preheat and then reduce heat to medium. For a charcoal grill, arrange medium-hot coals. Carefully transfer the dough circle directly onto the grill rack.
Cover and grill for about 2-3 minutes, or until the top is puffed and the bottom has beautiful grill marks and is crisp. Use a large spatula to carefully flip the crust over.
Grill the second side for another 2 minutes.
Remove the crust from the grill and place it on a baking sheet or large cutting board.
Working quickly, spread the tomato mixture evenly over the grilled crust.
Sprinkle generously with the shredded vegan mozzarella and vegan Parmesan.
Slide the pizza back onto the grill.
Cover and grill for about 3-5 minutes more, or until the crust is golden brown and the vegan cheeses are perfectly melted and bubbly.
Carefully remove the pizza from the grill, let it rest for a minute, then slice and serve immediately.
Don't overload the toppings. A thin, even layer ensures the crust stays crisp and doesn't become soggy.
Keep an eye on the heat.
Grill temperatures can vary, so check the bottom of the crust frequently to prevent it from burning.
