01 - Preheat your 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 the entire surface of the beef chuck roast, pressing gently to adhere.
02 - Heat olive oil in a large Dutch oven or oven-proof pot over medium-high heat. Sear the seasoned roast on all sides until a deep brown crust forms, approximately 8 minutes total, turning as each side browns.
03 - Pour the beef broth and 1 cup of BBQ sauce over the seared roast. Cover the pot 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.
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, adding extra BBQ sauce as desired for additional flavor and moisture.
05 - 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.
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 desired, then place the top bun. Serve immediately.