Preheat grill to medium high heat. Season chicken with salt and pepper. Grill chicken for 6 minutes. Flip and brush with barbecue sauce. Grill 4 to 5 minutes then flip and brush with barbecue sauce.
Top each chicken breast with 2 tablespoons shredded cheddar or 1 slice of cheddar. Close lid and allow cheese to melt. Remove chicken from grill. Top each chicken breast with a few avocado slices.