Preheat your barbecue or indoor grill 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. Set 2 slices of cheese on each mushroom and heat through 1 to 2 minute, or until cheese is starting to melt.
Spread the buns with the pesto-flavoured mayonnaise. Top the bottom buns with arugula. Set on the mushrooms, tomatoes and bell pepper, and then drizzle those ingredients with balsamic creme, if using. Set on top buns and serve.
Note: Balsamic creme, also called balsamic reduction, is reduced balsamic vinegar. Look for bottles of it in our Deli Department or vinegar aisle.