01 - Preheat oven to 325°F. Position the rack in the lower-middle position for even heat distribution.
02 - Pat the lamb shoulder dry with paper towels. Rub evenly with olive oil, then season generously with kosher salt, black pepper, smoked paprika, and garlic powder, pressing the spices into all surfaces.
03 - Place the seasoned lamb in a Dutch oven or roasting pan. Pour the broth and apple cider vinegar around the lamb, avoiding the top surface. Cover tightly with a lid or heavy-duty aluminum foil.
04 - Roast for approximately 3 hours, checking near the end. The lamb is ready when it pulls apart effortlessly with a fork and reaches an internal temperature of around 200°F for optimal tenderness.
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 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 or connective tissue. Fold in half of the prepared Alabama white sauce to coat the lamb evenly.
08 - Lightly toast the slider buns. Optionally brush the cut sides with melted butter before toasting for added richness and a golden finish.
09 - Pile the sauced pulled lamb generously onto the bottom halves of the toasted buns. Top with a mound of slaw and drizzle with additional Alabama white sauce if desired. Cap with the top buns.
10 - Serve immediately while the lamb is warm and the buns are fresh. Arrange on a platter for gatherings.