01 - Preheat oven to 325°F. Position the rack in the lower third of the oven for even heat distribution.
02 - Pat the lamb shoulder dry with paper towels. Rub evenly with olive oil, then coat generously 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 base of the lamb, avoiding washing off the seasoning. Cover tightly with a lid or aluminum foil.
04 - Roast for approximately 3 hours, checking occasionally, until the lamb is fork-tender and pulls apart easily with minimal effort.
05 - While the lamb roasts, whisk together the 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 the 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. 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. Optionally brush the cut sides with melted butter before toasting 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 if desired. Cap with the top buns and serve immediately.