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 season generously with kosher salt, black pepper, smoked paprika, and garlic powder, pressing the spices into the meat 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, being careful not to wash off the seasoning. Cover tightly with a lid or heavy-duty aluminum foil.
04 - Roast for approximately 3 hours, checking occasionally. 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 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 - In a separate bowl, toss the shredded cabbage and carrot with mayonnaise, apple cider vinegar, sugar, salt, and pepper until evenly coated. Cover and refrigerate until serving time to allow the flavors to meld.
07 - Remove the lamb from the oven and transfer to a large bowl. Using two forks, shred the meat while discarding any excess fat or connective tissue. Pour half of the Alabama white sauce over the shredded lamb and toss to coat evenly.
08 - Lightly toast the slider buns in a skillet or under the broiler. Brush the cut sides with melted butter for extra richness, if desired.
09 - Pile the sauced pulled lamb generously onto the bottom halves of the toasted buns. Top with a spoonful of slaw and an extra drizzle of Alabama white sauce if desired. Cap with the top buns and serve immediately.