Toast the guajillo and ancho chiles in a dry skillet over medium heat for 2 minutes until fragrant. Transfer to a bowl, cover with hot water, and soak for 15 minutes until softened.

Drain the chiles and add to a blender with quartered onion, garlic, cumin, oregano, apple cider vinegar, salt, pepper, and 1 cup of beef broth. Blend until completely smooth.

Place beef chunks in a large pot or Dutch oven. Pour the chile sauce over the beef, add remaining 3 cups of beef broth, and bring to a boil. Reduce heat to low, cover, and simmer for 2.5-3 hours until beef is tender and shreds easily.

Remove beef from the pot and shred with two forks. Strain the consomé through a fine-mesh sieve into a bowl, discarding solids. Reserve consomé for dipping and return some to the shredded beef to keep it moist.

Heat a large skillet or griddle over medium-high heat. Dip each tortilla briefly in the top layer of consomé to coat with the red oil, then place on the hot skillet.

Immediately add shredded cheese to one half of the tortilla, top with birria beef, fold in half, and press down. Cook for 2-3 minutes per side until golden and crispy.

Serve tacos hot with bowls of warm consomé for dipping, garnished with diced onion, cilantro, and lime wedges.