Heat gas or charcoal grill. Carefully brush oil on grill rack. Place tomatoes, cut sides up, and mushroom caps on grill over medium-high heat. Cover grill; cook tomatoes 4 to 5 minutes, mushrooms 6 to 8 minutes, turning once, until tender. Cool; peel tomatoes and chop. Chop mushrooms; set aside. Cut bell peppers lengthwise into quarters; remove stems and seeds. Place peppers, skin sides down, on grill over medium heat. Cover grill; cook 10 minutes or until skin is blackened. Place peppers in bowl; cover tightly. Cool 5 minutes. Peel off blackened skin; rinse. Chop peppers; set aside.
On lightly floured surface, roll dough to 15x11-inch rectangle. Brush with 1 tablespoon of the oil. Place oiled side down on grill over medium heat. Cook 4 to 6 minutes, turning once and brushing with remaining 1 tablespoon oil, until lightly browned and crisp. Place crust on waxed paper-lined cookie sheet. Spread with Alfredo sauce; sprinkle with Italian seasoning. Top with grilled vegetables, olives and cheese. Return pizza to grill until cheese is melted. Sprinkle with basil and thyme.