Classic North Carolina Pulled Pork

Name: Classic North Carolina Pulled Pork
Ingredients: ½ boneless pork shoulder (about 4 pounds), trimmed but with a layer of fat left on; 2 tablespoons coarse salt; 5 medium cloves garlic, pressed through a garlic press or mashed to a paste; 2 tablespoons safflower oil; 1 tablespoon granulated sugar; 2 tablespoons packed light-brown sugar; 1 tablespoon sweet paprika; ¾ teaspoon freshly ground black pepper; ¼ teaspoon dry mustard; ¼ teaspoon dried thyme; 1 cup white vinegar; ⅓ cup ketchup; ⅓ cup packed light-brown sugar; 1 tablespoon coarse salt; ¾ teaspoon red-pepper flakes; 1 teaspoon freshly ground black pepper; Potato buns, coleslaw, thinly sliced seedless cucumber, and pickles, for serving