1Heat closed medium-size contact grill for 5 minutes.
2Place mushrooms on grill. Close grill; cook 4 to 5 minutes or until slightly softened. Remove mushrooms from grill. Place onion on grill. Close grill; cook 4 to 5 minutes or until slightly softened. Remove onion from grill.
3In small bowl, mix mayonnaise and vinaigrette; spread over bread slices. Top 4 bread slices with mushrooms, cheese, onion, bell pepper and basil. Top with remaining bread, mayonnaise sides down.
4Place 2 sandwiches on grill. Close grill; cook 2 to 3 minutes or until sandwiches are golden brown and toasted. Repeat with remaining 2 sandwiches.