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 roast on all sides until deeply browned, approximately 8 minutes total.
02 - Pour beef broth and 1 cup of BBQ sauce over the seared roast. 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 effortlessly.
03 - Remove the Dutch oven from the oven and transfer the beef to a large cutting board. Using two forks, shred the meat into bite-sized pieces. Return the shredded beef to the pot and toss with the pan juices, adding extra BBQ sauce as desired for additional flavor and moisture.
04 - In a small bowl, whisk together mayonnaise, apple cider vinegar, Dijon mustard, honey, salt, and pepper until smooth and well combined.
05 - In a large mixing bowl, combine the shredded green cabbage, red cabbage, carrots, and sliced red onion. Pour the dressing over the vegetables and toss thoroughly until evenly coated. Refrigerate until ready to serve to allow flavors to meld.
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.