Preparation

• Heat a gas grill with all burners on high. Clean and oil the grate.
• In a small bowl, mix the mayonnaise, Parmigiano, and 1 tsp. of the garlic; spread the mixture over the cut sides of each roll.
• In another small bowl, whisk the olive oil, vinegar, and the remaining 2 tsp. garlic. Brush the oil mixture over both sides of each mushroom cap and sprinkle generously with salt and pepper. Grill the mushrooms, gill side down, until grill-marked, about 3 minutes. Flip and continue to grill until cooked through, about 3 minutes more. Reduce the heat to low, flip the mushrooms again, and top each with a portion of cheese. Put the rolls, cut sides up, on the grill along with the mushrooms. Cover the grill and cook until the rolls are crisp and warm and the cheese melts, 2 to 3 minutes.
• Put the rolls and mushrooms on a platter. Arrange a few basil leaves on the bottom half of each roll. Sprinkle the tomatoes lightly with salt. Put the tomatoes and then the mushrooms on the basil leaves. Cap the sandwiches with the tops of the rolls and serve.