01 - Preheat oven to 300°F. Pat the beef dry with paper towels and season generously with salt and pepper. Heat olive oil in a large Dutch oven over medium-high heat. Sear beef on all sides until deeply browned, about 3-4 minutes per side. Remove beef and set aside.
02 - In the same pot, add sliced onion and minced garlic. Sauté until softened and fragrant, about 3 minutes. Add smoked paprika, brown sugar, cumin, and chili powder. Stir constantly for 1 minute to toast the spices.
03 - Return beef to the pot. Pour in beef broth, barbecue sauce, apple cider vinegar, and Worcestershire sauce. Bring to a simmer, then cover tightly. Transfer to the oven and cook for 4 hours, or until beef pulls apart easily with a fork.
04 - Remove beef from the cooking liquid. Using two forks, shred the meat, discarding any large pieces of excess fat. Return shredded beef to the pot and toss thoroughly with the sauce. Keep warm until serving.
05 - Combine shredded green and red cabbage with grated carrots in a large bowl. In a separate bowl, whisk together mayonnaise, Greek yogurt, apple cider vinegar, Dijon mustard, sugar, salt, and pepper. Pour dressing over vegetables and toss until evenly coated. Refrigerate until ready to serve.
06 - Lightly toast the buns if desired. Pile generous portions of warm pulled beef onto the bottom half of each bun. Top with a generous layer of coleslaw. Drizzle with additional barbecue sauce if preferred. Crown with the top bun and serve immediately.