Place pork shoulder on a large parchment-lined baking sheet. In a small bowl, combine brown sugar, 2 tablespoons salt, 1 teaspoon black pepper, paprika, dry mustard, cumin, and cayenne. Rub pork with spice mixture, being sure to cover all sides. Cover and refrigerate at least 2 hours and preferably overnight.
Preheat oven to 325 degrees.
Remove pork shoulder from refrigerator and bring to room temperature, about 30 minutes. Pat dry with paper towels. Heat a large Dutch-oven over medium-high heat and add vegetable oil. Add pork shoulder to Dutch-oven and sear, turning, until golden brown all over, about 4 minutes per side.
Add 2 cups of water to Dutch-oven and position pork shoulder fat-side up, and cover. Transfer to oven and cook until meat shreds easily with two forks, 5 to 6 hours. Baste with pan juices every hour.
When cooked, remove from Dutch-oven and shred meat with two forks. Serve meat on buns with barbecue sauce.