01 - Preheat the oven to 300°F. In a small bowl, combine the smoked paprika, brown sugar, garlic powder, onion powder, black pepper, salt, and cayenne pepper. Rub the spice mixture evenly over all surfaces of the beef chuck roast, pressing it in with your hands to ensure good adhesion.
02 - Heat olive oil in a large Dutch oven or oven-safe pot over medium-high heat. Once the oil is shimmering, place the seasoned roast in the pot and sear on all sides until deeply browned, approximately 8 minutes total, turning every 2 minutes.
03 - Pour the beef broth and 1 cup of BBQ sauce over the seared roast. Cover the pot tightly with its lid and transfer to the preheated oven. Braise for approximately 3 hours, or until the beef is fork-tender and shreds apart with minimal effort.
04 - Remove the pot from the oven and transfer the beef to a cutting board. Using two forks, shred the meat, discarding any large pieces of fat. Return the shredded beef to the pot and toss with the pan juices. Add extra BBQ sauce if desired for additional flavor and moisture.
05 - In a large mixing bowl, combine the shredded green cabbage, red cabbage, carrots, and 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.
06 - 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 heaping scoop of crunchy coleslaw. Drizzle with extra BBQ sauce if you like, then cap with the top bun. Serve immediately.