Preheat the barbecue for medium heat and lightly oil the grate. Whisk together the balsamic vinegar, olive oil and garlic in a small bowl.
Arrange the portobello mushrooms gill-side up on a baking tray. Brush the mushrooms with some of the vinegar mixture and allow to marinate for 3 to 5 minutes.
Place the marinated mushrooms on the preheated barbecue gill-side down. Cook mushrooms until tender while brushing both sides of the mushrooms with the remaining marinade, about 4 minutes on each side.
Mix the mayonnaise, Dijon mustard, lemon juice and basil in a small bowl. Butter the toasted rolls then spread with the mayonnaise mixture. Divide the mushrooms, lettuce and tomato slices evenly to make 6 sandwiches.