Prepare the chiles: Toast guajillo, ancho, and chipotle chiles in a dry skillet over medium heat for 1-2 minutes until fragrant. Soak in 2 cups hot water for 15 minutes to soften.
Blend the sauce: Drain chiles and blend with onion, garlic, oregano, cumin, and 1 cup beef broth until smooth. Season with salt and pepper.
Sear the meat: Season beef or goat with salt and pepper. Heat 2 tbsp oil in a large Dutch oven over medium-high heat. Sear the meat for 3-4 minutes per side until browned. Remove and set aside.
Cook the birria: In the same pot, add chile sauce and cook for 2 minutes. Return meat, add remaining broth, cinnamon stick, and bay leaves. Bring to a simmer, cover, and cook on low for 3-4 hours until meat is tender.
Shred the meat: Remove meat, shred with two forks, and reserve consomé. Skim excess fat if desired.
Assemble tacos: Dip a corn tortilla in consomé, place on a hot skillet with a drizzle of oil, and add shredded cheese and meat. Fold tortilla and cook 2-3 minutes per side until crispy.
Serve: Garnish with cilantro, diced onion, and lime. Serve with consomé for dipping.
