Rinse pork roast under cold running water. In large oven bag or airtight container, combine apple cider, water, Worcestershire sauce, brown sugar, and salt. Mix until sugar and salt have dissolved. Add pork, cover, and refrigerate (overnight is best).When you're ready to cook your pork, preheat grill to 225 degrees F. If you have a meat injector, take some of the brine and inject it into the roast (I don't have one so I didn't do this, but it would have been awesome). Remove the roast from the brine mixture but save it for the cooking process.Mix up the one spoon dry rub and generously cover your roast using all of it.Put the roast on the grill with the fat side up and cook for 3 hours. After each full hour, use a silicon brush to mop the brine liquid onto the roast to keep it moist. You can discard the brine after the first 3 hours of roasting.Use an aluminium foil pan or make a boat out of heavy duty aluminum foil and place the roast on top. Increase the temperature to 250 degrees F and roast until the internal temperature reaches 150 degrees F, about 5-6 hours total.Transfer the roast to a cutting board and allow to rest for at least 20 minutes. Transfer any of the cooking juices in your foil to a measuring cup. When the meat is cool enough to handle, separate the tender meat from the bone and fat. Spoon off any fat that has risen to the top of the cooking liquid or use one of those measuring cups that pours from the bottom. Moisten shredded meat with cooking liquid.Serve with soft buns and Carolina mustard barbecue sauce along with some coleslaw.