01 - Preheat oven to 325°F.
02 - Pat lamb shoulder dry with paper towels. Rub evenly with olive oil, kosher salt, black pepper, smoked paprika, and garlic powder, coating all sides.
03 - Place seasoned lamb in a Dutch oven or roasting pan. Pour broth and apple cider vinegar around the lamb, avoiding the top surface. Cover tightly with a lid or aluminum foil.
04 - Roast for approximately 3 hours, or until the lamb is fork-tender and pulls apart easily with minimal effort.
05 - While the lamb roasts, whisk together mayonnaise, apple cider vinegar, prepared horseradish, lemon juice, Dijon mustard, sugar, cracked black pepper, garlic powder, onion powder, cayenne pepper, and salt in a mixing bowl until smooth and well combined. Refrigerate until ready to use.
06 - Toss shredded green cabbage, shredded carrot, mayonnaise, apple cider vinegar, sugar, salt, and pepper in a bowl until evenly coated. Refrigerate until serving.
07 - Remove lamb from the oven and transfer to a large bowl. Shred using two forks, discarding any excess fat. Fold in half of the Alabama white sauce, coating the pulled lamb evenly.
08 - Lightly toast slider buns, brushing with melted butter if desired.
09 - Pile the sauced pulled lamb onto the bottom halves of the toasted buns. Top with slaw and an extra drizzle of Alabama white sauce. Cap with the top buns and serve immediately.