Heat grill: When you're ready to cook your pork, preheat smoker grill to 225°F. If you have a meat injector, take some of the brine and inject it into the roast (this step is optional).
Prep pork roast: Remove the roast from the brine mixture and blot dry with paper towels. Discard the remaining brine. Cover all sides of roast with mustard (use gloves or a brush). Generously cover all sides of the roast with the dry rub.
Smoke: Put the roast on the grill with the fat side up and cook for 3 hours.
Increase temperature: Use an aluminum foil pan or make a boat out of heavy-duty aluminum foil and place the roast on top and place them back to the center of the smoker grill. Increase the temperature to 250°F and roast until the internal temperature reaches at least 195°F (continue cooking up to an internal temperature of 200°F for a softer and easier-to-pull-apart pork). The total cooking time for a roast this size can vary anywhere from 4-6 hours.
Rest: Wrap tightly in aluminum foil. Allow it to rest for at least 20 minutes up to a full hour. This will ensure the meat will easily shred.
Shred: After the rest period, transfer the roast to a cutting board and pour any of the residual cooking juices that accumulated in the foil to a measuring cup. When the meat is cool enough to handle, separate the tender meat from the bone and fat (discard the bone and fat).
Moisten meat: If any fat rose to the top of the cooking liquid, spoon it off and discard the fat. Alternatively, use a fat separator. Moisten the shredded meat with the remaining cooking liquid.
Serve: Keep the shredded meat warm in a slow cooker on the warm setting or in a covered saucepan over low heat. Serve with soft brioche buns and your favorite BBQ sauce along with your favorite sides for a complete meal!