Using a kitchen syringe (or a large syringe from the pharmacy if you don't have one), inject the cooled, strained brine all over the meat. Use the corners of the scored marks as a guide. Cover with plastic wrap and refrigerate for 12 to 24 hours.

Roast the pork shoulder using indirect heat and add a bit of smoke. Prepare your grill with an offset firebox, or if you don't have one, place the coals on one side of the grill and cook the meat on the opposite side, never directly over the coals. Once the coals are ready, add applewood chips for smoking.

Place a large pan (at least as big as the pork shoulder) under the grill grate to catch the drippings, and pour in about 8 1/2 cups water (2 liters) to keep the cooking environment moist. Set the pork shoulder on the grill grate, close the lid, and maintain the grill temperature between 210°F and 250°F (100°C to 120°C) for about 2 hours to give the meat a light smoke flavor.

Meanwhile, prepare the spray sauce by mixing the barbecue sauce with the cider and pouring it into a spray bottle. Spray the meat about once every hour during cooking.

After 2 hours, brush any remaining cider and barbecue sauce mixture over the meat. Place the pork shoulder in an aluminum tray and cover tightly with aluminum foil (or parchment paper, as long as it is well sealed; using a tray helps collect the juices). Return to the grill, cover, and cook at 300°F (150°C) for about 3 more hours, or until a meat thermometer reads an internal temperature of 185°F (85°C).

Once the internal temperature reaches 185°F (85°C), remove the meat from the grill and let it rest, still covered, for 1 more hour. After resting, the pork shoulder will be ready to serve, with a texture that's in between firm and pulled pork. Serve with the reduced juices from the tray.