Stir pesto and mayonnaise in small bowl to blend. Season with salt and pepper.
Prepare barbecue (medium-high heat). Grill rolls, cut side down, until lightly toasted. Transfer to plates. Spread grilled side of rolls with pesto mayonnaise.
Brush mushrooms on both sides with oil; sprinkle with salt and pepper. Grill, rounded side up, 4 minutes. Turn mushrooms over. Cover and grill until mushrooms are tender, about 4 minutes. Place 1 mushroom on each roll bottom. Top each with enough red pepper pieces to cover, then with arugula and cheese. Press roll tops over and serve.