01 - Preheat oven to 300°F. In a small bowl, combine smoked paprika, brown sugar, garlic powder, onion powder, black pepper, salt, and cayenne pepper. Rub the spice mixture evenly over the entire surface of the beef chuck roast. Heat olive oil in a large Dutch oven over medium-high heat. Sear the beef on all sides until deeply browned, approximately 8 minutes total.
02 - Pour beef broth and 1 cup of BBQ sauce over the seared beef. Cover the Dutch oven tightly with a lid and transfer to the preheated oven. Braise for approximately 3 hours, or until the beef is fork-tender and shreds apart easily.
03 - Remove the beef from the Dutch oven and transfer to a large cutting board. Shred the meat using two forks, discarding any large pieces of fat. Return the shredded beef to the pot and toss with the pan juices. Add extra BBQ sauce as desired for additional flavor and moisture.
04 - In a large mixing bowl, combine the shredded green cabbage, red cabbage, carrots, and thinly sliced red onion. In a separate small bowl, whisk together the mayonnaise, apple cider vinegar, Dijon mustard, honey, salt, and pepper until smooth. Pour the dressing over the vegetables and toss thoroughly to coat. Refrigerate until ready to serve.
05 - Lightly toast the sandwich buns if desired. Pile a generous portion of shredded BBQ beef onto the bottom half of each bun. Top with a hearty scoop of crunchy coleslaw. Drizzle with extra BBQ sauce if preferred, then place the top bun. Serve immediately.