Combine salt, brown sugar, and pepper. Rub over meat. Refrigerate seasoned pork overnight or while heating the grill.
Set up: Follow the manufacturer’s instructions to heat your charcoal or gas grill for cooking over indirect, low heat (200°F to 275°F). Plan for 4 1/2 to 6 hours of cooking time, about 1 1/2 to 2 hours per pound of meat. Check your grill owner’s manual for how to place a drip pan under the grill rack (some grills come with a drip pan) to catch the fat released during cooking to avoid flare-ups.
Place the pork on the area of your grill used for indirect cooking. Close the grill and roast pork, checking every 1 to 1 1/2 hours to adjust heat as needed and turn pork for even cooking. Cooking time will depend on your piece of meat, grill, and cooking temperature. Meat is done when the internal temperature at the thickest part reaches 195°F to 205°F, the crust is deep brown, and the meat is tender enough to pull apart with a fork.
Remove meat from grill and let rest for at least 20 to 30 minutes.
Shred meat or chop into bite-sized pieces. Drizzle barbecue sauce over pork and toss to blend.
Serve on halved brioche rolls with extra barbecue sauce on the side along with coleslaw and potato salad.
You can grill the meat a day ahead and reheat it to serve.
To Oven-Roast: Preheat oven to 250°F. Season meat as in step 1. Place on a rack in an oiled, shallow, heavy roasting pan. Roast for 1 1/2 to 2 hours per pound (4-6 hours total), turning meat once or twice, as needed for even cooking. Meat is done when the internal temperature at the thickest part reaches 195°F to 205°F, the crust is deep brown, and the meat is tender enough to pull apart with a fork. Follow steps 4-6 to shred and serve pork.