Prepare your smoker or pellet grill for 225 degrees F. Use cherry or pecan wood for this recipe.
Season the chuck roast with salt and pepper. Place it in the smoker, close the lid, and cook for an hour.
After an hour, it's time to add the short ribs. Season with salt and pepper, and place on the smoker with the chuck roast. Close the lid and cook both for an hour.
While the meat is on the smoker, trim off stems from the dried chilis and remove the seeds. Place the chilis and the tomato in a heat-safe bowl and pour boiling water over them. Cover bowl tightly with plastic wrap, and set aside for 15-20 minutes.
Heat oven to 325 degrees F.
Remove the chuck roast and short ribs from the smoker and place them in a large Dutch oven. You can cut the chuck roast into four equal sections if preferred. Set aside.
Heat oil in a skillet, and cook chopped onions over medium-low heat for 3 minutes until they take on a shiny appearance. Add one chopped shallot and garlic to the onions and cook for 1 minute.
Drain the chilis and add them to a blender. Cut the tomato in half and add it to the blender along with the onion-shallot mixture, beef broth, salt, pepper, and spices. Blend until smooth. Taste and adjust to your liking.
Pour the braising liquid over the meat, place the lid on the Dutch oven, and place it into the oven. Cook for 3 hours, turning the meat halfway through and add more broth or water as needed.
Test the meat for tenderness. It should shred easily. If not, place it back into the oven for 30 more minutes.
Remove the pot from the oven and let stand for 10 minutes.
Transfer the meat to a cutting board, but leave the sauce (consommé) in the pot.
Shred the meat and place it into a bowl.
Next, it's time to make your birria tacos. Gather your shredded cheese, chopped remaining shallot (or onion if preferred), and corn tortillas. This process goes quickly, so it's best to have all the taco components in place.
Heat a large skillet or griddle pan on your stovetop over medium-high heat. Add some oil and spread it around evenly.
Dip a corn tortilla in the consommé, transfer to a hot skillet, and top with cheese, shredded beef, and chopped shallot (or onion). Cook as you would a quesadilla. Fold it in half with a spatula. Wipe the skillet clean every 2-3 tacos to avoid burning. Add more oil as needed.
Serve on a platter topped with chopped cilantro and remaining chopped shallots, lime wedges, and warmed consommé on the side. Plan on 3 tacos per person.
Store leftover fixings and meat in separate airtight containers in your refrigerator for 3-4 days. Reheat meat in microwave or stove top in a pot.