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Preheat an electric smoker to 235 degrees F. Load the smoker with apple wood, cherry wood or hickory chips.
Evenly apply a thin layer of yellow mustard to the pork. Then sprinkle the rub all over the meat.
Place the roast in a disposable foil pan or directly on the smoker rack.
Cook for 2 hours. After 2 hours, brush the apple juice over the pork once per hour.
At the 4 hour mark, check the temperature of the meat. When the internal temperature is at 160 degrees F, brush with remaining apple juice or any drippings that have accumulated if you're cooking your meat in a pan. It can take around 5 hours to reach this temperature.
Wrap the meat in peach butcher paper or foil. Increase the smoker temperature to 250 degrees F.
Smoke the meat for another 3-4 hours or until the internal temperature registers at least 195 degrees F. Cook the meat to 205 degrees F if you prefer softer pulled pork.
Remove the meat from the smoker and let stand wrapped for 10 minutes and unwrapped for 5 minutes.
Shred the meat. Toss with BBQ sauce, then serve.