Prepare a gas or charcoal grill to high heat. Brush the grill with a bit of oil.
Brush eggplant, zucchini, and bell pepper with 2 tablespoons oil. Grill eggplant 4 minutes on each side or until tender; place in a bowl. Grill zucchini 3 minutes on each side or until tender; add to eggplant. Place pepper quarters, skin side down, on grill rack; grill 6 minutes or until blistered. Place peppers in a zip-top plastic bag; seal. Let stand 10 minutes. Peel peppers; add to vegetable mixture. Coarsely chop vegetables. Add garlic to vegetables; toss to combine.
Shape the dough into a circular shape while on a clean floured surface. Let the dough sit for 5 minutes and then re-form to make sure it's as big as you'd like. Place the pizza dough on a lightly floured rimless baking sheet, or pizza peel. Transfer the dough over to the grill, and slide the pizza dough directly onto the grill. Grill the dough for about 2 minutes with the lid closed. Using a pair of tongs, lift the lid, and flip the dough over onto the other side and cook for about 2 minutes more. The dough should be golden brown with grill marks on both sides.
Spread the Pizza Sauce evenly over top side of crust, leaving a 1/2-inch border. Arrange vegetable mixture evenly over sauce; sprinkle evenly with remaining 1/2 teaspoon salt and black pepper. Top with cheese. Return pizza to grill rack, and grill for 4 minutes or until thoroughly cooked. Sprinkle with mint and basil and serve.