Trim excess fat from top, leave bottom fat. Using knife, cut down the fat line on the end of the roast until it lays almost flat.
Combine apple juice and brown sugar until dissolved. Fill meat injector with mixture and inject meat moving across in 1-inch increments. Discard remaining injection liquid.
Combine rub ingredients and sprinkle liberally all over. Let meat rest for 30 minutes before cooking.
Preheat one grill zone, opposite from side you will be cooking on, to 250-275 degrees. Add 1 cup of smoker chips in aluminum foil pouch to direct heat side or directly in the coals if using charcoal.
Place meat on grill grate with fat cap up. Cook until internal meat temperature reaches 150 degrees.
Remove and place meat on two layers of heavy duty foil (30 inches long). Coat with honey and brown sugar until completely covered. Place butter on top of meat and wrap in foil, sealing all edges. Place in metal pan and return to grill.
Cook until internal meat temperature is 205-210 degrees. Remove and let rest in sealed foil for 1 hour.
Shred meat, removing unwanted fat and cover in barbecue sauce.
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