Heat a large pot or Dutch oven over medium heat. Add the dried guajillo and ancho chilies and toast for 1-2 minutes until fragrant.
Remove the chilies from the pot and transfer them to a bowl. Cover with hot water and let them soak for 15-20 minutes until softened.
In the same pot, add chopped onion and garlic. Sauté until softened and translucent, about 5 minutes.
Drain the soaked chilies and transfer them to a blender or food processor. Add the sautéed onions and garlic, ground cumin, dried oregano, smoked paprika, salt, and pepper. Blend until smooth, adding a splash of broth if needed to thin out the mixture.
Return the blended chili mixture to the pot. Add the beef or goat meat chunks and enough broth to cover the meat.
Bring the mixture to a boil, then reduce the heat to low and simmer, covered, for 2-3 hours until the meat is tender and the flavors have melded together.
Once the meat is tender, use a fork to shred it into smaller pieces. Continue to simmer for an additional 30 minutes to thicken the sauce.
Taste and adjust seasoning with salt and pepper if needed.
Serve the birria hot in bowls with warm corn tortillas on the side. Garnish with diced onions, cilantro, lime wedges, and sliced radishes if desired.