In large bowl, combine first 3 ingredients. Form into 6 patties.
Prepare grill to medium heat. Place patties and vegetables on grill. Grill, turning occasionally, until veggies are tender and browned and internal temperature of burgers reaches 165°F on an instant-read food thermometer, about 8-10 minutes. Brush burgers with ¼ cup Bayou Bourbon Glaze the last 2-3 minutes. Top with cheese.
Meanwhile, in small bowl, combine mayonnaise and remaining 1 tablespoon Bayou Bourbon Glaze.
Spread toasted buns with bayou mayonnaise. Top with lettuce, tomatoes, burgers and veggies.