In a small bowl, combine mayonnaise, pesto, mustard, 1/4 teaspoon salt and 1/4 teaspoon pepper; set aside.
Brush mushrooms caps with olive oil; sprinkle evenly with salt and pepper. Grill mushrooms, smooth side down, close the grill lid and grill for 6 to 8 minutes, or until tender. Toast the buns.
Spread the pesto mayo mixture evenly over both sides of the toasted buns. Divide arugula evenly over bottom halves of buns. Top with peppers. Place mushrooms caps over peppers. Cover with the top buns.