Plan ahead, this recipe brines overnight and cooks all day, but it’s well worth the time for these delicious flavors.In a large bowl, combine all of the brine ingredients with 2 cups water and bring to a simmer over medium heat. Simmer until the salt and sugar have dissolved. Remove from heat and add 2 cups ice and stir until dissolved. Let brine cool completely.Pour over the pork butt in a food-safe bucket or food storage container. Be sure the pork is fully submerged in the brine. Cover and refrigerate overnight.When ready to cook, set Traeger temperature to 225℉ and preheat, lid closed for 15 minutes. For optimal flavor, use Super Smoke if available.While the grill preheats, remove the pork roast from the brine and reserve the liquid. Using a meat injector, inject the pork with some of the remaining brine about every 2 inches across the entire roast. Pour the rest of the brine into a bowl and set aside. This will be used as your mop sauce during the cooking process.Generously season the roast on all sides with the Traeger Pork & Poultry Rub. Put the pork roast on the grill grate, fat-side up and smoke for 3 hours, mopping with the apple juice brine every hour after the first hour.Transfer pork roast to a disposable aluminum foil pan large enough to hold the roast.Increase the grill temperature to 250℉, and place the pan directly on the grill grate. Continue to cook for 6 to 8 additional hours or until an instant-read thermometer inserted into the thickest part, but not touching the bone, registers 204℉ to 206℉. If the pork starts to brown too much, cover it loosely with aluminum foil. Carefully transfer the pork roast to a cutting board and let it rest for at least 20 minutes.Pour the juices from the bottom of the pan into a gravy separator.With your hands, pull the pork into chunks. Discard the bone and any lumps of fat, including the cap. Pull each chunk into shreds, and transfer to a large mixing bowl. Season pulled pork with additional rub and moisten with the reserved pork juice.For the coleslaw: In a small bowl combine mayo, sour cream, vinegar, sugar, salt and pepper. In a separate medium bowl, combine the cabbage, carrot and onion. Pour mayo mixture over the cabbage mixture and stir to coat.Add Traeger 'Que BBQ Sauce to pulled pork and mix well, or serve the barbecue sauce on the side. Pile the pork mixture on the hamburger buns and serve with coleslaw. Enjoy!