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, 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 pouring directly over the meat. 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 braises, whisk together the mayonnaise, apple cider vinegar, horseradish, lemon juice, Dijon mustard, sugar, cracked black pepper, garlic powder, onion powder, cayenne pepper, and salt to taste in a mixing bowl until completely smooth. Refrigerate until ready to use.
06 - Toss the shredded cabbage and carrot with mayonnaise, apple cider vinegar, sugar, salt, and pepper in a bowl until evenly coated. 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 to thoroughly coat the pulled lamb.
08 - Lightly toast the slider buns, brushing with melted butter if desired, until golden on the cut sides.
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.