Preheat your indoor grill or barbecue to medium-high. Remove the stem from each mushroom. (If desired, stems can be saved and used to flavour a stock.) Brush the mushrooms with the oil; season with salt and pepper. Grill the mushrooms, cap-side down, for 2 minutes. Turn over and cook 2 minutes more, or until the mushrooms are tender.
Spread the buns with the pesto mayonnaise. Top the bottom buns with arugula. Set on the mushrooms, tomatoes, cheese and top buns and serve.