Preheat Smoker - Prepare your smoker for indirect cooking targeting 275 degrees Fahrenheit. We used apple wood for our cook, hickory, or oak work well also.
Season - Coat the trimmed Chuck Roast with olive oil and then apply the beef rub liberally to all sides.
Smoke - Place the seasoned Chuck Roast onto the smoker, and insert an instant read thermometer into the center of the Chuck Roast to track the internal temperature. Close the lid and smoke until the internal temperature of the roast is between 165 and 170 degrees Fahrenheit. This will take roughly 4 – 5 hours. Cook to the internal temperature, not time.
Braise - Once the internal temperature of the Chuck Roast is between 165 and 170 – remove the temperature probes and then place the Chuck Roast in a smoker safe dish with the braising liquid. Cover tightly with foil and place the probe back into the Chuck Roast. Place back on the smoker and continue smoking until the internal temperature of the Chuck Roast is between 200 and 210 degrees F or when another temperature probe inserts easily like through room temperature butter. This can take 2 – 3 additional hours.
Rest - Then remove the dish with the braising Chuck Roast and let rest for 30 minutes covered.
Pull - Using large forks or a pulling tool, gently separate the meat and pull in small pulls in the same dish. This incorporates the pulled Beef into the braising liquid. You may run into fat pockets, if you do, use your hands to separate the good meat and discard the fat or cartilage. Continue to pull until you like the texture.
Top - Place the pulled Beef onto the bun, and then top with the banana peppers and some coleslaw. If you made extra sauce, drizzle it over the top and then add the top of the bun. Serve and enjoy warm!