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Combine the brine ingredients in a large bowl and stir until the brown sugar and salt have dissolved. Put the pork into the brine, ensuring it is fully submerged. Leave the pork in the brine for at least 24 hours.
When ready to cook, start the Traeger grill on Smoke with the lid open until the fire is established (4 to 5 minutes).
Take the pork shoulder out of the brine and set it directly on the grill grate, fat-side up. Allow it to smoke for 3 hours, spraying it with apple juice every hour.
Meanwhile whisk together the ingredients for the sauce in a bowl and set aside.
After smoking the pork shoulder for 3 hours, turn the temperature up to 250 degrees F and roast for an additional 5 hours, or until an instant-read meat thermometer inserted in the thickest part, but not touching bone, registers 195 degrees F. When it's ready you should be able to give the bone a gentle twist and pull, and the bone should come out with relative ease.
After it reaches 195 degrees F carefully take the pork off the Traeger and wrap with foil and a towel; keep in a cooler or warm oven for 30 minutes to an hour.
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. Add in the barbacoa sauce and mix together thoroughly.
Serve with black beans, cilantro lime rice, avocado, and a few sprigs of fresh cilantro. Enjoy!