Preheat the oven to 320 F. Rehydrate the chiles: place the guajillo chiles, ancho chiles, onion, garlic, and tomatoes into a large pot, pour in the broth, and simmer for 15 mins. Transfer to a blender and blend until smooth.
Heat the oil in a large Dutch oven or oven-safe pot and brown the meat pieces on all sides, for about 6 minutes. Pour the sauce over the meat, and mix in the vinegar, cumin, coriander, oregano, bay leaf, salt, and pepper. Place the lid on and roast for 3 hours.
Remove the meat and transfer it to a bowl. Shred using two forks and mix in ¼ cup of the birria broth. Place the broth left in the slow cooker in a different bowl.
Make the tacos: heat 1 tsp oil in a medium-sized cast iron skillet over medium heat. Dip one tortilla into the birria broth and place on the hot skillet. Add 3 tbsp of shredded meat, one slice of Oaxaca cheese, and 1 tbsp chopped fresh cilantro to one side of the tortilla. Fold the tortilla and cook on both sides until golden and crispy, About 1 minute on each side. Add more oil if needed until all tortillas are done. Serve with your favorite taco toppings.