01 - Preheat the oven to 325°F.
02 - Pat the lamb shoulder dry with paper towels. Rub evenly with olive oil, then coat with kosher salt, black pepper, smoked paprika, and garlic powder, pressing the seasonings into the meat.
03 - Place the seasoned lamb into 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 aluminum foil.
04 - Roast in the oven 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, horseradish, lemon juice, Dijon mustard, sugar, cracked black pepper, garlic powder, onion powder, cayenne pepper, and salt to taste in a mixing bowl until smooth and well combined. 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. Refrigerate until serving to allow the flavors to meld.
07 - Remove the lamb from the oven and transfer it to a large bowl. Using two forks, shred the meat, discarding any excess fat. Pour half of the Alabama white sauce over the shredded lamb and mix until evenly coated.
08 - Lightly toast the slider buns. Brush with melted butter if desired for added richness.
09 - Pile the sauced pulled lamb generously onto the bottom halves of the toasted buns. Top with the prepared slaw and drizzle with additional Alabama white sauce if desired. Cap with the top buns.
10 - Serve the sliders immediately while the lamb is still warm.