Combine mayonnaise and relish; spread evenly over 4 bread slices. Top evenly with chicken, cheese, and arugula. Top with remaining bread slices. Coat both sides of sandwiches with cooking spray.
Place a nonstick skillet coated with cooking spray over medium-high heat. Add 2 sandwiches to pan; cook 2 to 3 minutes on each side or until lightly browned. Repeat with remaining sandwiches.