Prepare outdoor grill for direct grilling over medium heat. Brush mushrooms, onion, and squash with oil: sprinkle with Salt & Pepper
Grill vegetables, covered, 6 minutes or until tender and browned, turning once. Removed from grill and separate onion rings; set aside. Reduce heat on grill to medium-low.
Cover large cookie sheet with fil; spray with nonstick cooking spray. Stretch dough into 10” by 14” rectangle. Place on cookie sheet.
Lift dough and foil; place, dough side down, on grill, gently peeling off foil. Cover; cook 2 minutes or until bottom is crisp. Turn crust over. Quickly top with tomatoes, vegetables, and cheese. Cover; cook 2 minutes longer or until bottom is crisp. Slide onto cutting board; garnish with basil.