Toast the guajillo and ancho chiles in a dry skillet over medium heat for about 2-3 minutes until fragrant. Remove from the skillet and place in a bowl.
In the same skillet, add a cup of water and bring it to a simmer. Pour the hot water over the toasted chiles and let them soak for about 15 minutes to soften.
Drain the softened chiles and place them in a blender. Add the quartered onion, garlic, cumin, oregano, black pepper, and salt. Blend until you achieve a smooth paste, adding a little beef broth if necessary.
In a large Dutch oven, heat the vegetable oil over medium-high heat. Add the beef chunks and sear them on all sides until browned. Remove the beef and set aside.
In the same pot, add the blended chile paste and cook for 5 minutes, stirring frequently, until fragrant.
Return the seared beef to the pot, pour in the beef broth, and add the bay leaves. Bring to a boil, then reduce the heat to low, cover, and let it simmer for 3-4 hours or until the beef is tender and easily shredded.
Once cooked, remove the beef from the pot and shred it using two forks. Return the shredded beef to the pot and stir well.
Serve the birria hot with corn tortillas, lime wedges, and chopped cilantro on the side. Pour the broth (consommé) into bowls for dipping.