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, pressing the seasonings into the meat.
03 - Place the seasoned lamb in a Dutch oven or roasting pan. Pour the broth and apple cider vinegar around the lamb, being careful not to wash off the seasoning. Cover tightly with a lid or heavy-duty aluminum foil.
04 - Roast in the oven for approximately 3 hours, or until the lamb is fork-tender and pulls apart easily with minimal effort.
05 - While the lamb braises, whisk together mayonnaise, apple cider vinegar, prepared horseradish, lemon juice, Dijon mustard, sugar, cracked black pepper, garlic powder, onion powder, cayenne pepper, and salt to taste in a mixing bowl until smooth and well combined. 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 coated. Refrigerate until serving to allow flavors to meld.
07 - Remove the lamb from the oven and transfer to a large bowl. Using two forks, shred the meat, discarding any excess fat. Fold in half of the Alabama white sauce, coating the pulled lamb evenly.
08 - Lightly toast the slider buns. Brush with melted butter if desired for extra richness.
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 and serve immediately.