There is nothing like cooking pulled pork in a smoker. The BBQ pulled pork flavors cannot be faked or replicated making it well worth the time and effort to cook properly.
Trim the pork to remove any tags, residual bone shards, and excessive fat.
Pork Rub
Rub the pork with the mustard, then sprinkle with the pork rub to evenly coat. Bring up to room temperature, about 1 hour.
Preheat a smoker to 120 degrees C (250 degrees F). Add 3 large hickory or similar wood chunks.
Smoke the pork for 3 hours. This can be done directly on the grates in the smoker, or the pork can be placed in a disposable half-pan.
Pork Wrapping
Combine the apple juice, honey, and brown sugar. Cut the butter into pats.
Place the pork on top of a large piece of foil. Pour over the apple juice mixture and butter. Wrap with the foil. Then wrap tightly with a second layer of foil.
Return the pork to the smoker and continue to cook for 3 additional hours. Check the pork for doneness by probing for the desired texture and about a 90 degrees C (195 degrees F) internal temperature. However also take into account the resistance when probing the pork, it should be soft and give little resistance. If it is hard to get the probe in cook longer until soft.
Pork Glaze
Combine the BBQ sauce, pork rub, and apple juice.
Remove the pork and unwrap carefully as there will be steam and liquid at the bottom. Carefully roll the foil down to create a self-made tray, alternatively, you can pour the juice and move the pork to a new half-pan. Glaze the pork then return to the smoker. Cook for an additional 15 minutes to let the glaze set.
To Serve
Remove the pork from the smoker and rest for 30 to 90 minutes before pulling.
Place heatproof gloves on and pull the pork into small pieces, and remove any large chunks of fat. Pour a small amount of the liquid from cooking in along with some of the leftover glaze. Taste and add more glaze if needed. Serve while still hot.