01 - Preheat oven to 325°F.
02 - Pat the lamb shoulder dry with paper towels. Rub evenly with olive oil, kosher salt, black pepper, smoked paprika, and garlic powder, ensuring full coverage on all sides.
03 - Place the seasoned lamb in a Dutch oven or roasting pan. Pour the broth and apple cider vinegar around the lamb, avoiding washing off the seasoning. 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 completely smooth. Refrigerate until ready to use.
06 - Toss shredded cabbage and carrot with mayonnaise, apple cider vinegar, sugar, salt, and pepper in a bowl until evenly coated. Refrigerate until serving to allow flavors to meld.
07 - Remove the 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 the slider buns, brushing with melted butter if desired, until golden on the cut sides.
09 - Pile the sauced pulled lamb generously onto the bottom halves of the toasted buns. Top with slaw and an extra drizzle of Alabama white sauce. Cap with the top buns.
10 - Serve immediately while the lamb is warm and the buns are fresh.