Preheat grill to medium-high heat.
Brush mushrooms caps with olive oil; sprinkle evenly with salt and pepper. Grill mushrooms, smooth side down, covered with grill lid, for 6 to 8 minutes, or until tender.
In a small bowl, combine mayonnaise, pesto, mustard, 1/4 teaspoon salt and 1/4 teaspoon pepper. Spread mixture evenly over cut sides of buns. Divide arugula evenly over bottom halves of buns. Top each evenly with peppers. Place mushrooms caps over peppers. Cover with tops of buns.