01 - Preheat oven to 325°F.
02 - Pat the lamb shoulder dry with paper towels. Rub evenly with olive oil, then season with kosher salt, black pepper, smoked paprika, and garlic powder, coating all sides.
03 - Place the seasoned lamb into a Dutch oven or roasting pan. Pour the broth and apple cider vinegar around the lamb, avoiding pouring directly over the meat. Cover tightly with a lid or aluminum foil.
04 - Roast in the center of the oven for approximately 3 hours, or until the lamb pulls apart easily with a fork and reaches melt-in-your-mouth tenderness.
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. Using two forks, shred the meat, 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 and warm.
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.