Heat gas or charcoal grill. In small bowl, mix mayonnaise spread ingredients. Refrigerate. With small metal spoon, scrape underside of mushroom caps to remove dark gills.
Spray mushrooms, bell peppers and onion slices with cooking spray. Place on grill over medium heat. Cover grill; cook 7 to 10 minutes, turning occasionally, until bell peppers and onion are crisp-tender. Remove vegetables from grill; let stand until cool enough to handle.
Meanwhile, cut focaccia in half horizontally to form 2 rounds. Spread mayonnaise spread evenly on cut side of bottom half.
Slice mushrooms; arrange over mayonnaise spread. Cut bell peppers into thin strips; layer over mushrooms. Separate onion slices into rings; place over peppers. Top with tomato and cheese slices. Cover with top half of focaccia.
If desired, wrap sandwich in foil; place on grill. Cook 1 to 2 minutes or until cheese is melted. To serve, cut into 6 wedges.