Prepare a charcoal grill for indirect heat using lump hardwood charcoal to 225°F-250ºF. Place a drip pan under the grate where the pork will be positioned. Place a hickory log on top of the coals and make sure you have a good smoke before adding the pork.
Oil the cooking grates and place the shoulder, fat side down, on the grill, and cook until the internal temperature reaches 130ºF, about 3-3 ½ hours, adding hickory logs as needed and maintaining the temperature of the coals. Turn the shoulder fat side up and cook until the internal temperature reaches 160ºF, about 3 ½-4 hours, adding hickory logs as needed, and maintaining the temperature of the coals. Spray the shoulder with the mop mixture every 30 minutes.
Tear off two pieces of heavy-duty aluminum foil that will be large enough to completely wrap the pork shoulder, and place, perpendicular from each other, over a baking sheet. Remove the shoulder from the grill and place, fat side up, on the foil. Spray, one more time, with the mop. One sheet at a time, wrap the shoulder tightly. Return the wrapped shoulder to the grill, and cook until the internal temperature reaches 190ºF, about 3 ½ hours, adding Hickory logs as needed and maintaining the temperature of the coals.
Meanwhile, line a cooler with thick bath towels to insulate the inside of the cooler. Remove the pork from the grill. Keep the aluminum foil wrap in place and wrap the pork in plastic wrap. Place the pork in a disposable pan and place it in the cooler. Cover the top of the pork with another towel, close the lid and allow the meat to rest for 1 hour.
At this point the meat will be very tender and ready to shred. Remove the pork from the cooler and place it on a cutting board. Carefully unwrap the pork, reserving any juices. Shred the pork with forks or bear claws. Transfer the pork to a serving dish and toss with the reserved juices. The shredded pork will keep in the refrigerator for up to 3 days. leftovers can be frozen in an airtight bags or containers or up to 6 months. Enjoy!