Whisk together the brown sugar, paprika, chili powder, garlic powder, onion powder, thyme, cumin, cayenne, salt and pepper.
Pat the chicken breasts dry with a paper towel and arrange on a plate or baking sheet.
Sprinkle the rub on the chicken and coat both sides evenly.
Drizzle the olive oil over the chicken and place on the grill.
Cook the chicken for 3-4 minutes or until it easily releases from the grill and is slightly charred.
Flip the chicken and brush the top with barbecue sauce to coat. Cook for an additional 4-5 minutes.
Flip the chicken again and brush with the barbecue sauce to coat. Turn the heat to low and cook for an additional 3-4 minutes or until the chicken registers 150˚F on an instant read thermometer. (The internal temperature will continue to rise to a safe 165˚F while resting. Pulling it off the grill a little early ensures the chicken will be juicy and not overcooked.)
Coat the chicken with a little more sauce and transfer to a platter. Cover with foil and rest for 5 minutes.
Serve immediately with more sauce, if desired.
You can prep the chicken with the dry rub for up to 24 hours ahead of time.Store in the refrigerator until ready to grill. Allow the chicken to rest for 30 minutes at room temperature before grilling.