To prepare the balsamic glaze, combine the balsamic vinegar and honey in a saucepan over medium low heat for about 20 minutes or until thickened. Set aside.
Brush the mushrooms with olive oil and place on a sheet pan. In a skillet over medium heat, add 1-2 T olive oil. When hot, add sliced onion, salt and pepper and cook until caramelized, about 15-20 minutes. Set aside. In a small bowl, combine the mayonnaise and pesto. Cover and place in refrigerator until ready to use.
Prepare roasted pepper on a grill or char on a gas stove top. Cook until skin is thickened and blistered all over. Transfer pepper to a small bowl and cover tightly with plastic wrap for about 10-15 minutes. Take pepper out of bowl, remove skin, seeds, ribs, slice and place back in bowl. Cook mushrooms on a medium grill turning until heated through, about 5-7 minutes. Set aside. Toast the buns on the grill and transfer to a platter.
To assemble sandwich, spread each set of the rolls with 2 T of pesto mayonnaise. Add a grilled mushroom cap, cheese, onion, roasted pepper and arugula. Drizzle with the balsamic glaze followed by the top roll. Serves 4.