Combine mayonnaise and minced basil; cover and chill.
Combine oil and next 3 ingredients in a large bowl. Add bell pepper, eggplant, onion, and zucchini to dressing mixture; toss vegetable mixture well to coat.
Place bell pepper and eggplant on grill rack coated with cooking spray; grill 6 minutes. Add onion and zucchini; grill 5 minutes. Turn vegetables often, baste with dressing, and cook just until tender. Remove from grill, and keep warm.
Place bread on grill, cut side down, and grill 2 minutes. Turn bread, cut side up, and place half of cheese slices on bottom half of bread. Grill 1 minute or until cheese begins to melt.
Spread mayonnaise mixture over top half of bread. Top bottom half with vegetables, remaining cheese slices, and top half of bread. Cut crosswise into fourths.