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, ensuring full coverage on 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, 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. 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 combined. 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 to coat the lamb evenly.
08 - Lightly toast slider buns, brushing with melted butter if desired.
09 - Pile sauced pulled lamb onto bottom halves of toasted buns. Top with slaw and drizzle with additional Alabama white sauce. Cap with top buns and serve immediately.