Prep the grill. Preheat pellet grill or smoker to 250℉.
Prep the chuck roast. Add a thin layer of yellow mustard to all sides of the chuck roast as a binder. Follow with a thin layer of Worcestershire sauce on all sides. About ¼ cup total of each. Sprinkle a liberal amount of Texas Rub and 'Que That (or your favorite barbecue seasoning) on all sides of the chuck roast.
Smoke the chuck roast. Place the chuck roast directly on the grill grates and smoke for about 4-5 hours, or until the internal temp is about 160-165℉.
Prepare the braising liquid. To a half-sized aluminum foil pan, add beef stock, apple cider vinegar, Worcestershire sauce, hot sauce, barbecue sauce, and finely diced onion. Mix to combine.
Braise the chuck roast on the grill. Place the smoked chuck roast in the aluminum foil pan, and spoon the braising liquid on top. Cover pan tightly with aluminum foil and cook on the grill at 325℉ until internal temp of chuck roast reaches about 208-210℉ and roast can be shredded (about 1.5 - 2 hours).
Shred the BBQ beef. Pour pan juices/braising liquid into a separate bowl. Use two forks to shred the smoked chuck roast in the aluminum foil pan, discarding any large pieces of fat as you shred. Skim the excess fat off the braising liquid. Pour about 1 cup of the skimmed braising liquid back in the foil pan with the shredded beef. Mix to combine. Pour about ½ - 1 cup of barbecue sauce on top of the shredded beef.
Cook BBQ beef on the grill to thicken the sauce. Put the shredded BBQ beef back on the pellet grill or smoker uncovered, and cook for an additional 30-45 minutes at 325℉. Remove from the grill, and stir to redistribute the sauces.
Serve your shredded smoked BBQ beef sandwiches. Serve shredded BBQ beef with hamburger buns or Texas Toast of your choice. Top with coleslaw, thinly sliced onions, dill pickles, or your favorite barbecue sandwich toppings.