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, ensuring full coverage 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, avoiding washing off the seasoning. Cover tightly with a lid or aluminum foil.
04 - Roast in the oven for approximately 3 hours, or until the lamb is fork-tender and pulls apart effortlessly.
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 smooth and well combined. Refrigerate until ready to use.
06 - In a separate bowl, toss shredded green cabbage and carrot with mayonnaise, apple cider vinegar, sugar, salt, and pepper until evenly coated. Refrigerate until serving.
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 lamb thoroughly.
08 - Lightly toast the slider buns. Brush with melted butter if desired 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. Cap with the top buns and serve immediately.