Prep The Beef. Place the chuck roast and short ribs on a baking sheet and generously season it with salt all around. Transfer the baking sheet to the refrigerator and chill for 30 minutes.
Prep The Peppers. Add the ancho and guajillo chillies to a pot with boiling water. Cover with lid and let them soak for about 5 minutes until they begin to soften. Transfer the chillies to a cutting board and carefully remove the seeds and stems. Set aside. I recommend using gloves.
Toast The Spices. Heat a skillet over medium heat. Add the cinnamon stick, black peppercorns, and cloves. Toast for about 2 minutes or until aromatic. Transfer them to a spice grinder or blender including the bay leaves and blend into a fine powder.
Sear The Beef. In a large Dutch oven heat the oil over medium-high heat. Add the chilled meat to the Dutch oven and sear on all sides until golden brown, working in batches. Transfer the meat to a plate and set aside.
Cook Onion And Garlic. In the same Dutch oven add the onion and cook for about 5 minutes or until onion softens. Add the garlic and cook for another 30 seconds until aromatic.
Add Remaining Ingredients And Cook. To the Dutch oven, add the tomato paste, the ground spices, oregano, lemon juice and chiles. Add the meat back to the pot, pour in the water and bring to a boil. Reduce the heat to medium-low, cover and simmer for 30 minutes. Taste for salt and adjust as necessary.
Prep The Oven. Preheat the oven to 300°F.
Blend The Chiles. Carefully remove the chiles from the pot using tongs and place them in a blender with a couple cups of the broth from the Dutch oven. Blend until smooth. Pour this pepper mixture back into the Dutch oven and stir well.
Cook Beef In The Oven. Cover the Dutch oven and transfer it to the preheated oven. Cook for 2 to 3 hours or until the meat shreds easily with 2 forks.
Shred The Beef. Transfer the meat from the Dutch oven using tongs to a large bowl. Reserve the broth (this is called consommé). Shred the beef using 2 forks. Set aside. Strain the consommé through a sieve.
Fry Tacos. Heat about 1 tablespoon of oil in a large skillet. Dip a tortilla into the consommé then place it in the skillet. Top with some shredded Oaxaca cheese, and shredded beef. Fold the taco in half then flip it over to cook until the cheese melts, another 1 to 2 minutes.
Serve And Enjoy. Serve tacos with leftover consommé topped with some red onion, cilantro and a wedge of lime.