Preheat an outdoor grill 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 tray or baking sheet. 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 grill, gill-side down. Grill mushrooms until tender, 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 kaiser rolls, then spread with the mayonnaise mixture. Divide the mushrooms, lettuce, and tomato slices evenly to make 6 sandwiches.
