Trim excess skin and fat from chicken.
Combine salt and brown sugar in cold water in a large bowl and stir until dissolved. Refrigerate two to four hours.
Preheat one grill zone, opposite from side you will be cooking on, to 300-325 degrees.*, Remove chicken from brine and rinse with cold water.
Cut butter in ¼ inch slices and place in foil pan. Place wire rack in pan and chicken on the rack, bone side up. Cover entire chicken with rub and cover tightly with foil.
Place foil covered pan on grill. No wood chips are needed yet. Cook 1 to 1½ hours, checking meat temperature at 1 hour. Once chicken reaches 185 to 195 degrees, remove from grill.
Warm sauce and dip each thigh into the sauce and place on a rack. Serve or for a little smoky flavor, add wood chips or smoke pouch to grill.
Once smoke is generated, place chicken (still on rack) on the grate and smoke for 15 minutes at 300 to 325 degrees. Do not over-smoke.