In a zippered plastic bag, mix 1/4 cup olive oil and 2 tablespoons balsamic vinegar. Place the onion rings in a bag, shake the bag, fasten and leave the onion to marinate for 30 minutes.
Meanwhile, in a small bowl, combine the garlic, basil, and the remaining olive oil. Lubricate the mushroom caps on all sides with this mixture, set aside.
In a small saucepan, combine the honey and the remaining vinegar, bring to a boil over medium heat, cook, stirring, for about 5 minutes, until lightly thickened. Remove the saucepan from the stove, let cool slightly.
Grease the grill grate with vegetable oil and turn on the grill to preheat to medium temperature. Remove the onion from the marinade, dry on a paper towel. Place the onion rings and mushroom caps on the grill rack, and fry on all sides for about 10-12 minutes.
Cut the buns in half, put lettuce, tomato, cheese, mushroom cap, and onion in turn on one half, pour over everything with honey sauce and cover with the other half of the bun. Thus, collect all the burgers.