Step 1

In a large heat-proof bowl, add ancho and guajillo chiles. Pour the boiling water over the chiles. Soak for 45 minutes. Remove the chiles and place in a blender. Blend until smooth, adding some of the soaking water as needed to blend into a smooth but thick sauce. Set aside.

Step 2

Sprinkle the chuck roast and short ribs with 1 tablespoon of salt.

Step 3

In a large Dutch oven, heat the oil over medium-high heat. Working in batches, sear the meat until browned on all sides, about 2 minutes per side. Remove from the pot.

Step 4

Reduce the heat to medium and add the onion and carrot to the pot. Cook, stirring frequently, until the vegetables are tender and browned, about 7 to 10 minutes. Add the tomatoes, garlic, cumin, oregano, and remaining 1 teaspoon salt. Cook, stirring constantly for 2 minutes more, using the spoon to scrape any bits off the bottom of the pot. Stir in the chicken broth and pureed chiles. Add the meat, bay leaves, and cinnamon stick, and bring the mixture to a boil over medium heat.

Step 5

Reduce the heat to low. Cook until the meat is fork-tender or reaches 205°F to 210°F when an instant-read thermometer is inserted into the thickest portion of the meat, about 3 to 3 ½ hours. Remove from the heat.

Step 6

Remove the meat, and discard the cinnamon stick and bay leaves. Shred the meat with two forks, and discard the bones.

Step 7

Skim the fat off the top of the birria broth and reserve in a heat-proof bowl. Reserve the broth in the pot for serving.

Step 8

When ready to eat, heat 1 tablespoon of vegetable oil in a 12-inch cast-iron skillet over medium heat, and swirl to coat the bottom of the skillet.

Step 9

Working in batches, place a tortilla on the surface of reserved broth in the pot and flip to coat. Place into the preheated skillet. Cook for 30 seconds to 1 minute, or just until starting to brown. Flip the tortilla. Sprinkle each tortilla with 2 to 3 tablespoons of cheese then top half with about ¼ cup of shredded meat. Fold the other half of the tortilla in half, covering the meat.

Step 10

Cook until bottoms start to get crispy, about 30 seconds to 1 minute. Flip the tacos and fry the other side until they start to crisp, about 30 seconds to 1 minute. Remove from the pan and repeat, adding additional oil as needed. Serve with onion, cilantro, limes wedges, and warm reserved birria broth for dipping.