COMBINE pesto and mayonnaise in medium bowl; stir well.
PREHEAT grill to medium-high. Brush both sides of Portobello mushrooms with oil; season with salt and pepper. Grill on both sides for about 4 minutes. Remove from grill and keep warm. Toast cut sides of rolls on grill just until light golden; remove.
SPREAD bottoms of each roll with about 2 tablespoons pesto mayo. Top with mushroom, cheese slice, red onion, roasted red pepper and small handful of arugula. Top with top of roll.