Preheat smoker to 250 degrees Fahrenheit (F) using a fruit wood like apple or cherry.
Coat Chuck Roast with olive oil and liberally apply the beef rub. Insert a temperature probe into the center of the roast and place on the smoker. Smoke for 3 hours, or until the internal temperature of the roast is 165 degrees F.
To Make the Pot Roast Stew
When the internal temperature of the Chuck Roast is at 155 degrees F, start the stew. Place a Dutch oven on the stovetop over medium heat. Add olive oil, onions, carrots, and celery and sauté for 10 minutes or until soft and translucent but not browned. Add garlic and cook another 1 minute, stirring.
Add the wine and bring to a simmer for 5 minutes. Add the tomatoes, stock, rosemary, thyme, chili pepper flakes, salt, and pepper bring back to a simmer. At about this time the roast will be at or close to 165 degrees F. When this happens, place the roast into the braising liquid and cover. Transfer the stew pot, with the chuck roast in it, onto the smoker.
Increase the smoker to 325 degrees F and continue to braise for 2 hours or until the internal temperature of the roast reaches between 200 - 210 degrees F, or until you probe the chuck with a fork and the fork goes into the chuck like butter. Remove from smoker for final step to thicken the braise.
Remove the roast and the herbs from the pot and set aside in a large bowl or a large cutting board.
Combine the flour and butter in a small bowl and add to the braising liquid. Bring the braising liquid to a simmer and stir for 5 minutes, as you stir it will thicken.
Cut the roast into eight pieces and serve with the liquid over mashed potatoes or on its own. Alternatively you can shred the beef (shredding the beef is actually my favorite way to serve this dish).
If you make the braise and the roast is not at 165, you can still transfer the roast to the Dutch oven and finish in the grill. Or you can keep the braising liquid on the lowest possible simmer until the 165 is reached and then add the roast.