To prepare the slaw: mix together mayo, vinegar, sugar and salt in a large bowl. Add broccoli slaw to bowl and mix to combine. Refrigerate until ready to eat. It tastes better if made a few hours ahead of time.
Preheat grill to medium-high heat. Season both sides of chicken with salt and pepper. Grill chicken on one side for 6 minutes, flip and brush with barbecue sauce. Cook another 6 minutes, flip and brush second side with barbecue sauce. Cook 2 minutes to caramelize sauce, flip and cook another 2 minutes. Remove from grill when cooked through.
To assemble chicken sandwiches, add chicken and slaw to buns.