Special equipment: A charcoal or wood-fired grill or smoker; a meat injector
Light a fire in the smoker or grill. Allow the fire to burn for about 2 hours until the temperature in the cooker is 225 degrees F.
Meanwhile, remove the pork shoulder from the refrigerator and it rest at room temperature for 1 hour.
Bring the chicken broth to a boil in a large pot. Add 2 cups of the rub, then add the sugar, barbecue sauce, and Worcestershire sauce and simmer until dissolved. Using a handheld mixer or whisk, gradually incorporate the melted butter and olive oil. Mix until the broth and fat emulsify then set the marinade aside to cool.
Rub the pork shoulder all over with the remaining cup of dry rub then set the meat aside for 20 minutes. Measure the marinade, and then add enough apple juice to achieve a 3 to 1 marinade-to-juice ratio. Inject the marinade and juice mixture into the pork repeatedly until the meat is saturated and can absorb no more.
Cook the pork shoulder at 225 degrees F in the grill or smoker (adding more coal or wood as necessary to maintain the temperature). After about 4 hours the pork should have turned a rich red-brown. Remove the meat from the cooker, wrap it in aluminum foil, and then return it, fat side down, to the grill or smoker. At this point begin checking the temperature of the meat with a thermometer every hour until it reaches an internal temperature of between 180 and 190 degrees F. at the center. (This should take about 4 more hours; the bones will pull clean while the meat stays firm.)
Glaze the shoulder with barbecue sauce, mixed with a little corn syrup for extra shine if you like. Using a large fork, pull the meat from the bones and serve.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.