Pour 3/4 cup warm water in the bowl of a stand mixer with dough hook attached. Weigh or lightly spoon flour into dry measuring cups and spoons; level with a knife. Add flour to 3/4 cup water; mix until combined. Cover and let stand 20 minutes. Combine remaining 1/4 cup water and yeast in a small bowl; let stand 5 minutes or until bubbly. Add yeast mixture, 4 teaspoons oil, and 1/2 teaspoon salt to flour mixture; mix 5 minutes or until a soft dough forms. Place dough in a large bowl coated with cooking spray; cover surface of dough with plastic wrap lightly coated with cooking spray. Refrigerate 24 hours.
Prepare grill to high.
Remove dough from refrigerator. Let stand, covered, 1 hour or until dough comes to room temperature. Punch dough down. Press dough out to a 12-inch oval on a lightly floured baking sheet sprinkled with cornmeal. Crimp edges to form a 1/2-inch border. Cover dough loosely with plastic wrap.
Brush eggplant, zucchini, and bell pepper with remaining 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.
Place pizza dough, cornmeal side up, on grill rack coated with cooking spray, and grill for 4 minutes or until blistered. Turn dough over; grill 3 minutes. Remove from grill. Spread Basic 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. Cut pizza into 10 pieces; sprinkle with mint and thyme.