Heat grill. In small bowl, combine all spread ingredients; mix well. Refrigerate. With small metal spoon, scrape underside of mushroom caps to remove dark gills.
When ready to grill, spray mushrooms, bell peppers and onion slices with nonstick cooking spray. Place on gas grill over medium heat or on charcoal grill 4 to 6 inches from medium coals. Cook 7 to 10 minutes or until bell peppers and onion are crisp-tender, turning occasionally. 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.