01 - Preheat oven to 325°F.
02 - Pat the lamb shoulder dry with paper towels. Rub evenly with olive oil, kosher salt, black pepper, smoked paprika, and garlic powder, coating all sides.
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 aluminum foil.
04 - Roast for approximately 3 hours, or 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 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.
07 - Remove the lamb from the oven and transfer to a large bowl. Shred the meat using two forks, discarding any excess fat. Fold in half of the Alabama white sauce to coat the lamb evenly.
08 - Lightly toast the slider buns, brushing with melted butter if desired.
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.
10 - Serve the sliders immediately while warm.