In a large pot, heat vegetable oil over medium-high heat. Sear the chuck roast chunks until browned on all sides. Remove meat and set aside. Add chopped onion and garlic to the pot, sauté until softened, about 5 minutes. Add guajillo and ancho chiles, cumin, and oregano; toast for 1 minute until fragrant. Pour in beef broth and apple cider vinegar, bring to a simmer.

Return the seared beef to the pot. Cover and reduce heat to low. Simmer for at least 2-3 hours, or until the beef is fork-tender. Alternatively, cook in a pressure cooker for 45-60 minutes. Once cooked, remove the beef and shred it using two forks. Strain the cooking liquid (consomé) through a fine-mesh sieve into a separate bowl; discard solids. Skim off excess fat if desired.

Heat a large griddle or cast-iron skillet over medium heat. Dip each corn tortilla into the consomé, coating both sides. Place the dipped tortilla on the hot griddle. Sprinkle a generous amount of shredded Oaxaca cheese over one half of the tortilla. Add a spoonful of shredded birria meat on top of the cheese.

Once the cheese starts to melt and the bottom of the tortilla is crispy, fold the other half of the tortilla over to create a taco. Press down gently with a spatula. Cook until both sides are golden brown and crispy, and the cheese is fully melted and slightly browned. Repeat with remaining tortillas and filling.

Serve the quesabirria tacos immediately with small bowls of the warm consomé for dipping. Garnish with fresh chopped cilantro and diced white onion.