Start by trimming any excess fat from the Beef Chuck and place it in a large pot. Add one peeled Onion (halved), several cloves of Garlic, a Bay Leaf, and a pinch of Salt. Pour in enough Water to just cover the meat and bring to a gentle boil, then reduce to a simmer and cook until the beef is fork-tender, about 2 to 2½ hours.
While the beef cooks, toast the stems-removed Guajillo and Ancho chiles in a dry skillet over medium heat just until fragrant, about 20–30 seconds per side; be careful not to burn them.
Transfer the toasted chiles to a bowl, cover with hot water, and let soften for 15 minutes, then drain and place in a blender with a peeled Tomato, one peeled Onion (use a second onion here), a couple cloves of Garlic, a pinch of Mexican Oregano, a pinch of Ground Cumin, and a few grinds of Black Pepper. Blend until smooth, adding a splash of the chile soaking liquid if needed to reach a saucy consistency.
Strain the blended chile-tomato sauce through a sieve if you prefer a smoother texture, then taste and adjust seasoning with Salt and a tablespoon of Vinegar to brighten the flavors.
When the beef is done, remove it from the pot and reserve about a cup of the cooking liquid (this is your Beef Broth). Shred the meat finely with two forks or by hand.
Heat a few tablespoons of Oil in a skillet over medium heat. Add the shredded beef and pour in the prepared chile-tomato sauce and enough reserved Beef Broth to keep the mixture moist but not soupy. Simmer together for 10–15 minutes so the meat absorbs the sauce, tasting and adding more Salt or Vinegar if needed.
Warm the Tortilla in a dry skillet or wrapped in a towel in the microwave until pliable. If using, spread a thin smear of Refried Beans across the center of each tortilla as a base.
Spoon a generous portion of the deshebrada (shredded beef in chile sauce) onto each warmed Tortilla, fold in the sides and roll tightly to form burritos.
Serve the burritos hot with extra sauce on the side, pickled onions, or salsa of your choice. Leftover deshebrada keeps well refrigerated for several days and can be reheated gently with a splash of broth.
