Grill tomatoes over high heat, uncovered, turning on all sides, until skins are somewhat blackened and split and tomatoes are soft, about 5 to 6 minutes. As soon as tomatoes are cool enough to handle, halve them and squeeze out the watery liquid. Purée tomatoes in a blender or food processor with tomato paste, 2 tablespoons basil, extra-virgin olive oil, garlic-flavored olive oil (optional) and salt.
Place tomato sauce, mozzarella, red pepper flakes (optional), 1/2 cup chopped basil and a spoon within easy reach of the grill. Place a baking sheet on a convenient work surface near the grill and have a large spatula or pizza peel ready.
Place first crust on the grill, laying it directly on the cooking grate over high heat. Grill, uncovered, for 1 to 1 1/2 minutes, until crust becomes firm but is still flexible (don't worry about any bubbles that form on the crust, as they will be flattened out when crust is turned over).
Using a spatula or peel, flip crust onto the baking sheet, cooked side up. Immediately spoon on half of the tomato sauce and sprinkle with half of the cheese (and, if you wish, a bit of the red pepper flakes). Quickly return pizza to grill (without the baking sheet), uncooked side down. Arrange pie so that half of it is over high heat and the other half is over medium-low heat. Cook pizza for another 3 to 4 minutes, rotating it in quarter turns every 30 to 45 seconds (this may sound awkward but becomes second nature very quickly). Using the spatula to lift the edge slightly, check the bottom during the last minute or two, rotating a bit faster or slower if needed to get a uniformly brown, crisp crust. Scatter the basil over the top shortly before removing the pizza from the grill. Slice pizza into wedges and serve immediately. Repeat process for second pizza.
Nutrition information per each slice of pizza:
Calories340Fat16 gSodium550 mg