Prepare the Blackberry Glaze: Stir together 4 cups blackberries, 2 cups white balsamic vinegar, 2 cups water, 1 cup granulated sugar, and 6 thyme sprigs in a medium saucepan. Bring to a boil over medium-high; reduce heat to medium-low, and simmer until blackberries have broken down, about 15 minutes. Remove from heat; cool 10 minutes. Remove and discard thyme sprigs. Transfer mixture to a blender. Remove center piece of blender lid (to allow steam to escape); secure blender lid on blender. Place a clean towel over opening in lid (to avoid splatters). Process until smooth. Pour pureed mixture through a fine wire-mesh strainer; discard solids. Stir in 1 1/2 tsp. kosher salt and 3/4 tsp. black pepper.

Prepare the Grilled Pork Loin: Stir together rosemary, oil, and garlic; rub mixture all over pork, and sprinkle with salt and pepper. Let stand at room temperature about 30 minutes.

Preheat a charcoal grill to medium-high (about 450°F). When briquettes are covered in gray ash, push them to one side, and scatter hickory wood chunks over coals. Place pork, fat side down, on oiled grate directly over coals, and grill, covered, until browned, about 10 minutes. Turn pork, and move to side without coals. Grill pork, covered, 30 minutes, turning twice. Brush with 1/3 cup Blackberry Glaze, and grill, covered, until a thermometer inserted in thickest portion of pork (away from bone) registers 145°F, about 25 minutes, turning and brushing with glaze twice. Remove pork from grill, and let rest about 10 minutes. Slice into chops, and serve immediately.