Generously season the meat with salt and pepper on all sides, and heat the olive oil in a large pot or Dutch oven over medium-high heat.
Working in 2 batches, add the meat and sear on all sides until browned. Remove the pot from the heat, add the seared meat back into the pot, and set aside.
While the meat is searing, add the guajillo chiles, ancho chiles, arbol chiles, tomatoes, onion, cinnamon stick, bay leaves, and peppercorns to a medium pot. Cover completely with water and bring to a boil over high heat. Reduce the heat to low, cover, and cook for 10 minutes.
Using a slotted spoon, transfer the softened chiles and all the other ingredients to a large blender.
Add 1 cup of the chile-soaked cooking water, the beef broth, white vinegar, garlic, cumin, oregano, and cloves. Blend on high for a few minutes until completely smooth. (You may have to do this in 2 batches if your blender isn’t big enough.)
Strain the blended sauce through a fine mesh strainer into the pot with the seared meat. Discard any solids left behind.
Stir the meat and chile sauce together to combine and bring to a boil over high heat. Reduce the heat to low, cover, and simmer for 3 to 3 ½ hours until the meat is fall-apart tender.
Transfer the meat to a large bowl and shred it with 2 forks. Add the meat back into the consomé and serve as a stew in bowls with diced onion and chopped cilantro or as tacos in corn tortillas with shredded Oaxaca cheese, cilantro, and diced onions.
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Notes
Mexican cinnamon: This cinnamon is also known as Ceylon cinnamon. It’s brittle, can be easily broken into small pieces, and is actually blended into the sauce. It’s readily available in most Hispanic grocery stores. If you can’t find it near you, I recommend using a regular cinnamon stick and removing it before blending or using 2 teaspoons of ground cinnamon.
Mexican oregano: If you don’t have Mexican oregano, you can use regular oregano instead.
Spice level: The spice level of this recipe is somewhere between medium and hot. For a true medium spice, use only 3 chiles de arbol. For mild spice, don’t use any chiles de arbol.
To make this in the Instant Pot: Use the 'Saute' function to sear the meat, then add in the blended chile sauce. Close the lid and pressure cook on high for 60 minutes using the manual setting. Let the pressure release naturally for 10 minutes before manually releasing any remaining pressure.
To make birria in the slow cooker: Brown and sear the meat in a pot or skillet. Add the seared meat and blended chile sauce to the slow cooker, cover, and cook on low for 7-8 hours or on high for 5 hours.