Preheat a grill to medium-high heat. In a medium bowl, stir the chopped tomatoes with 1 tablespoon olive oil, the vinegar, 1/2 teaspoon salt and 1/4 teaspoon pepper. Toss well and let stand to become juicy and soft while you make the pizza.

Divide the dough into 2 pieces. Stretch each piece of dough into a large rectangle on a lightly floured work surface using your hands or a rolling pin, about 10-inches long by 6-inches wide. Transfer to the back of a large baking sheet using your hands and reshape the rectangle, if necessary. In a small bowl, mix the remaining 3 tablespoons oil with the grated garlic and lightly brush the top of each piece of dough.

Working with one piece of dough at a time, place the dough, oiled-side down, on the grill until the dough puffs and the bottom is lightly charred with deep grill marks, 2 to 3 minutes. Flip the dough, using heatproof tongs and a spatula. Brush what is now the top of the pizza with half the oil and sprinkle with half the Parmesan. Grill until the cheese is melted and the pizza is crisped, about 2 minutes. Transfer to a large cutting board. Repeat with the second dough rectangle.

Cut the grilled pizza into 2-inch wide strips and cut each in half crosswise. Mash the tomato mixture, using a potato masher, to form a chunky sauce and stir in the basil. Serve the flatbread strips with the tomato sauce on the side.