In a saucepan, bring 4 cups of water to a boil. Remove from heat and add the dried guajillo and ancho chilies. Let them soak for about 15 minutes until softened.
Drain the chilies and transfer them to a blender. Add the chopped onion, minced garlic, ground cumin, dried oregano, salt, and pepper. Blend until you have a smooth sauce.
Cut the beef chuck roast into smaller pieces and place them in a bowl. Pour the chili sauce over the beef and toss to coat the meat evenly. Allow it to marinate for at least 30 minutes or longer for more flavor.
Preheat your air fryer to 375°F (190°C). Place the marinated beef pieces in the air fryer basket in a single layer. Air fry for about 8-10 minutes or until the beef is cooked and slightly crispy on the edges. Remove and set aside.
While the beef is cooking, cook the frozen French fries in the air fryer according to the package instructions until they are crispy and golden.
Once the fries are cooked, remove them from the air fryer and place them on a serving plate or platter.
Sprinkle the shredded mozzarella cheese over the hot fries, allowing it to melt slightly.
Place the cooked beef Birria on top of the cheesy fries. Garnish with chopped fresh cilantro and serve with lime wedges on the side.