Remove stems and seeds from the guajillo and ancho chiles. Toast them briefly in a dry skillet over medium heat for 1–2 minutes until fragrant, then transfer to a bowl and cover with hot water to soften.

Hydrate and blend the paste

Drain the softened chiles, reserving some soaking liquid. In a blender, combine the hydrated chiles, garlic, white onion, tomatoes, chipotle peppers in adobo, cumin, oregano, coriander, salt, and pepper. Blend until smooth, adding soaking liquid a little at a time to achieve a thick, glossy paste.

Sear the beef

Season the short ribs with salt and pepper. In a large Dutch oven, heat olive oil over medium-high heat and sear the beef on all sides until deeply browned. Remove and set aside.

Saute aromatics

In the same pot, add the chopped onion and garlic from the paste and sauté until translucent and lightly browned.

Deglaze and build the braise

Deglaze the pot with a portion of the chipotle sauce, scraping up browned bits. Return the beef to the pot and stir in the chile paste. Add beef stock and bay leaves.

Braise the beef

Bring to a simmer, then cover and braise on low heat or in a 325°F (165°C) oven for 3–4 hours, until the meat is fork-tender. Shred beef into large bite-sized pieces and discard any bones if used.

Make the consommé

Strain the braising liquid to remove solids. Return the liquid to the pot and simmer until reduced to about 4 cups. Stir in fresh lime juice and adjust seasoning with salt to taste.

Pickle the onions

Combine the sliced red onion with white vinegar, sugar, and salt in a jar. Let marinate for 20–30 minutes (or longer for stronger flavor). Drain before serving.

Prepare tortillas and cheese

Warm a skillet over medium heat. Lightly melt Oaxaca cheese on each tortilla, or simply place a small amount of cheese on the tortilla and allow it to melt as you add the beef later.

Assemble and serve

Fill each tortilla with seasoned, shredded birria beef and melted cheese. Top with pickled onions and chopped cilantro. Serve with a cup of warm consommé for dipping and lime wedges on the side.