Step 1

Prepare a grill for indirect heat. If using a charcoal grill, build the hot coals on one side only. If using a gas grill, heat one side only to medium-high heat.

Step 2

Place a heavy-bottomed Dutch oven over the direct heat side of the grill. To the dry pan add the ancho chiles, chiles de árbol and guajillo chiles and toast until slightly charred and fragrant, 2 to 3 minutes. Remove and set aside. To the same Dutch oven, add a drizzle of olive oil. Season the chuck roast and short ribs with salt and pepper. Add to the hot oil and deeply brown on all sides, 2 to 3 minutes per side. Add the garlic, carrot, onion, cumin, oregano and chili powder. Add the toasted chiles, then pour in the chicken stock and add the bay leaves. Bring to a simmer, move to the indirect heat side of the grill, close the lid of the grill and simmer for 30 minutes, allowing the chiles to re-hydrate.

Step 3

Using tongs, pull the chiles out and add to a blender along with 1 to 2 cups of the broth. Cover the lid with a kitchen towel to catch splatters. Puree until smooth then pour the mixture back into the pan.

Step 4

Add 2 cups of water to the blender to rinse it out and pour into the Dutch oven. Cover the grill with a lid and continue to braise until tender, another 1 to 1 1/2 hours. Try to maintain a temperature of 275 to 300 degrees F. Skim any fat off the top of the pot and save to use to toast the tortillas. Pull the meat from the liquid and shred into smaller pieces using 2 forks. Strain the broth through a fine-mesh sieve and discard the solids. Keep the broth warm.

Step 5

Place a cast-iron skillet or braiser over the direct heat side of the grill. Using a pastry brush, brush some of the reserved fat all over the skillet. Using tongs, dip 2 of the tortillas into the broth, soaking both sides, and place in the pan. Immediately sprinkle the surface of each tortilla with about 1/4 cup of the cheese and 1/4 cup of the shredded braised beef. Allow the cheese to begin to melt and the tortillas to toast, about 1 minute. Fold each tortilla in half then toast on one side until golden brown and crisp, about 1 minute. Flip to the other side and allow to crisp, about 1 minute. Repeat with the remaining tortillas, cheese and meat.

Step 6

To serve: Ladle the hot broth into small bowls. Garnish each bowl of broth with finely diced onion and cilantro. Serve alongside the tacos for dipping with some lime wedges on the side.