01 - Preheat oven to 325°F.
02 - Pat the lamb shoulder dry with paper towels. Rub evenly with olive oil, then coat with kosher salt, black pepper, smoked paprika, and garlic powder 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 green cabbage and carrot with mayonnaise, apple cider vinegar, sugar, salt, and pepper in a bowl until evenly combined. Refrigerate until serving time.
07 - Remove the lamb from the oven and transfer to a large bowl. Shred the meat using two forks, discarding any excess fat. Fold in half of the Alabama white sauce, coating the pulled lamb thoroughly.
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 a portion of slaw and an extra drizzle of Alabama white sauce. Cap with the top buns and serve immediately.