Begin by preparing the dried chiles. Remove stems and seeds from the guajillo, pasilla, and ancho chiles. Place them in a dry skillet over medium heat for a couple of minutes until fragrant.
Transfer the toasted chiles to a blender. Add 1 cup of beef broth, garlic, onion, cumin, oregano, thyme, black pepper, and salt. Blend until smooth to form a sauce.
In a large Dutch oven, heat a little oil over medium-high heat. Add the goat meat and sear on all sides until browned, then remove from the pot and set aside.
Pour the blended sauce into the pot and bring to a simmer, scraping the bottom of the pot to incorporate any browned bits.
Return the seared meat to the pot, then add the remaining beef broth, apple cider vinegar, and bay leaves. Bring everything to a boil.
Reduce the heat to low, cover the pot, and let it simmer for about 3 hours or until the meat is tender and falls apart easily.
Once cooked, remove the meat from the pot, shred it with two forks, and return it to the pot. Adjust salt to taste.
Serve the birria hot, garnished with fresh cilantro, chopped onion, and lime wedges. Pair it with warm corn tortillas on the side for dipping.