In a resealable plastic bag, combine 1/4 cup oil and 2 tablespoons vinegar; add onion slices. Seal bag and turn to coat; marinate for 30 minutes. Meanwhile, in a small bowl, combine the garlic, basil and remaining oil. Brush over both sides of mushroom caps; set aside.
In a small saucepan, bring honey and remaining vinegar to a boil; cook for 5 minutes or until thickened. Set aside.
Drain and discard marinade from onions. Using long-handled tongs, moisten a paper towel with cooking oil and lightly coat the grill rack. Grill onions and mushrooms over medium heat or broil 4 in. from the heat for 10-12 minutes or until tender, turning frequently.
Serve on rolls with lettuce, tomato and mozzarella. Drizzle with reserved balsamic-honey sauce.