Heat gas grill to medium-high or charcoal grill until coals are ash white.
Combine olive oil, rosemary, garlic, salt and pepper in bowl. Reserve 3 tablespoons oil mixture in another bowl; set aside.
Add bell pepper, zucchini, mushrooms and tomato to oil mixture in large bowl; toss to coat. Place into grill basket; grill, stirring occasionally, 8-10 minutes or until vegetables are tender. Keep warm.
Turn gas grill to low or move coals to 1 side in charcoal grill.
Brush both sides of pizza crust with 1 1/2 tablespoons reserved oil mixture. Place pizza crust onto grill (opposite heat on charcoal grill), top-side down. Grill 2-3 minutes or until light golden brown. Turn crust over; immediately place cheese onto crust, covering crust. Close lid; grill 6-10 minutes or until bottom of crust is golden brown and cheese is beginning to melt.
Spread grilled vegetables over cheese on pizza; remove from grill. Top with basil, if desired. Cut into 8 pieces; serve immediately.