Heat a dry skillet or comal over medium heat. Toast the guajillo, ancho, and chile de arbol chilies for 1-2 minutes per side, just until fragrant and slightly pliable. Be careful not to burn them, as this will make them bitter.

Transfer the toasted chilies to a heatproof bowl. Add the halved Roma tomatoes, quartered white onion, and peeled garlic cloves to the same bowl. Pour 2 cups of hot water over everything, ensuring the chilies are fully submerged. Let them soak for 20-30 minutes until softened.

Once softened, transfer the chilies, tomatoes, onion, garlic, and about 1 cup of the soaking liquid to a blender. Blend until completely smooth. If needed, add a little more soaking liquid (or fresh water) to achieve a smooth, pourable consistency. Strain the adobo sauce through a fine-mesh sieve into a bowl, pressing firmly on the solids to extract all liquid. Discard the remaining pulp.

Pat the beef chuck roast and short ribs dry with paper towels. Season generously with 1 tsp Kosher salt and 1/2 tsp black pepper.

Heat 2 tablespoons of olive oil in a large Dutch oven over medium-high heat until shimmering. Sear the beef in batches, ensuring not to overcrowd the pot, until deeply browned on all sides, about 3-4 minutes per side. Remove the seared beef and set aside.

In the same Dutch oven, add the roughly chopped large white onion and smashed garlic. Sauté for 3-5 minutes until softened and fragrant, scraping up any browned bits from the bottom of the pot.

Return the seared beef to the Dutch oven. Pour the strained chili adobo sauce over the beef. Add the bay leaves, cumin seeds, Mexican oregano, whole cloves, cinnamon stick, apple cider vinegar, beef broth, 2 cups of water, and the remaining 1 tsp Kosher salt and 1/2 tsp black pepper. Stir to combine.

Bring the mixture to a simmer, then reduce the heat to low, cover tightly, and cook for 3.5 to 4 hours (or until the beef is fall-apart tender). Alternatively, cook in a slow cooker on low for 6-8 hours or high for 3-4 hours.

Carefully remove the cooked beef from the Dutch oven and place it in a large bowl. Using two forks, shred the beef into bite-sized pieces. Discard any bones or large pieces of fat. Skim some fat from the top of the cooking liquid if desired, but leave some for flavor.

Strain the remaining cooking liquid (consomé) through a fine-mesh sieve into a clean pot or heatproof bowl. Press on any solids to extract maximum liquid. Taste the consomé and adjust seasoning if necessary. Keep the consomé warm.

Heat a large skillet or comal over medium-high heat. Lightly dip each corn tortilla into the consomé, ensuring both sides are coated with the flavorful liquid and some of the reddish fat. Place the dipped tortilla on the hot skillet.

Sprinkle a generous amount of shredded Oaxaca cheese over one half of the tortilla. Add a spoonful of the shredded birria beef on top of the cheese. Once the cheese begins to melt and the tortilla crisps slightly, fold the tortilla in half to create a taco.

Continue cooking for 1-2 minutes per side, pressing gently with a spatula, until the tortilla is golden brown and crispy, and the cheese is fully melted and gooey.

Repeat with the remaining tortillas, beef, and cheese.

Serve the hot birria tacos immediately with small bowls of the warm consomé for dipping. Garnish with chopped fresh cilantro, finely diced white onion, and lime wedges. Thinly sliced radishes are an optional, refreshing addition.