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.