Toast the guajillo and ancho chiles in a dry skillet over medium heat for about 1 minute per side. Transfer them to a bowl, cover with boiling water, and soak for 15 minutes until softened.
Drain the chiles and add them to a blender along with the fire roasted tomatoes, garlic, onion, cumin, oregano, cloves, salt, and pepper. Blend until very smooth.
Place the beef chuck pieces in a pot, pour the chile sauce and beef broth over them. Bring to a boil, then reduce to a low simmer, cover, and cook for 2 to 2.5 hours until the meat shreds easily with a fork. Shred the beef and return it to the broth.
Dip each corn tortilla into the birria broth so it soaks up flavor and color. Place on a plate and fill one half with shredded beef and a generous pinch of cheese, then fold over.
Preheat the air fryer to 380F. Lightly spray the basket and arrange the filled tacos in a single layer. Air fry for 5-6 minutes, flipping once halfway, until the tortillas are deeply crispy and the cheese is melted.
Serve the tacos with a small bowl of the warm birria broth for dipping, topped with cilantro and a squeeze of lime.