Preheat. Preheat your smoker to 275 degrees F.
Season. Season the chuck roast on all sides with Fiesta Rub.
Make the braising liquid. In a 12-inch cast iron skillet or aluminum pan, combine 4 cups of the beef stock, peppers, garlic cloves, onion, and cilantro.
Smoke. Place the pan of braising liquid on the lowest grates of your smoker, set the seasoned chuck roast on a rack above the pan of braising liquid. With this set up, you'll maintain moisture in your smoker, as well as catch any drippings from your roast. Close the lid and smoke your roast for 4-5 hours, or until the internal temperature reads 165- 175 degrees F.
Braise. Move the roast down into the pan of braising liquid, close the lid, and cook for an additional 2-3 hours, or until your roast reads 210 degrees F internal temperature and shreds easily with a fork.
Shred. Remove the roast and shred into small pieces. Strain the peppers and onions from your braising liquid. It should have reduced significantly while cooking. Whisk in the remaining 2 cups hot beef stock to reconstitute your consomme and keep warm.
Make the tacos. Prepare your workstation for cooking the birria tacos. Preheat a cast iron skillet or flat top over medium-high heat and oil lightly. Dip your tortilla in your consomme liquid and set it directly on the oiled pan. Cook for 1-2 minutes before flipping. Top the tortilla with cheese and layer in your shredded birria. Fold the tortilla over and cook, flipping occasionally, until the cheese is melted and the tortilla is browned and crisp. Repeat with the remaining tortillas, cheese, and birria.
Enjoy. Serve your tacos with fresh lime, cilantro, and diced onion alongside a bowl of your hot consomme (topped with more cilantro and onion). Dip your taco into that consomme before each bite and enjoy!