Spray the inside halves of the top and bottom of the bun with cooking spray. Place on a heated flat griddle, oiled-side down. Grill until golden brown.
Arrange the bottom half of the bun on a warmed serving plate, browned side up. Spread 1 oz. of drained pickled red onions on top of the bun. Place the warmed chicken on top of the onions. Place a piece of muenster cheese on top of the chicken. Place under a heated salamander to melt the cheese.
Spread 1 oz. of drained pickled carrots out on top of the cheese. Place ¼ oz. of baby arugula on top of the carrots. Drizzle 1 Tbsp. of cilantro mayonnaise on top of the arugula. Lean the top of the bun against the sandwich.