Begin by boiling the beef chuck roast in a large pot of water for about 30 minutes. Remove from heat, let it cool, and shred the meat once it's cool enough to handle.
In the same pot, add the beef broth, dried guajillo chilies, dried ancho chilies, garlic, onion, ground cumin, oregano, salt, and black pepper. Bring to a boil, then reduce heat and simmer for 15-20 minutes.
Once the chilies are softened, transfer the mixture to a blender and blend until smooth to create the sauce.
Preheat your oven to 350°F (175°C).
In a skillet, lightly sauté the corn tortillas for about 30 seconds on each side until warm and pliable.
Fill each tortilla with shredded birria meat and a sprinkle of cheese. Roll the tortillas up and place them seam-side down in a greased baking dish.
Pour the blended sauce over the top of the enchiladas. Sprinkle with the remaining cheese.
Cover the baking dish with aluminum foil and bake for 25 minutes. Remove the foil and bake for an additional 10 minutes until the cheese is melted and bubbly.
Let the enchiladas cool for a few minutes before serving. Garnish with fresh cilantro, lime wedges, and optional sliced radishes.