To make rub, place first 5 ingredients in a bowl and combine with your fingers, breaking up lumps in sugar. (You'll need 1 1/2-2 tablespoons rub for this recipe. The excess can be stored in a sealed jar for several weeks.)
Sprinkle rub over chicken on both sides, rubbing into the meat with your fingers. Arrange chicken in a baking dish. Cover with foil, and refrigerate 20 minutes.
To make basting mixture, combine oil and smoke seasoning in a small bowl, stirring with a whisk.
Spread the coals in an even layer to make a single-zone fire. (On a gas grill, you'd simply light the appropriate burner.) Preheat to high. Clean and oil grate.
Arrange chicken diagonal to bars of grate, and grill 4-6 minutes total per side or until browned and thoroughly heated; begin basting chicken after first 2 minutes of cooking, and baste several times. (If you're feeling ambitious, rotate each breast a quarter-turn after the first 2 minutes to create hatch marks.) Insert an instant-read meat thermometer into thick end of breast through the side--the temperature should be about 16
Transfer chicken to a platter or plates; serve with barbecue sauce.
