For the beef: Combine the ancho, New Mexico and de arbol chiles, tomatoes, onion, garlic and beef broth in a large pot. Place over medium heat and bring to a gentle simmer. Continue to simmer for 12 to 14 minutes while you sear the beef.
Heat the oil in a large Dutch oven over medium-high heat. Season the chuck roast with salt and pepper. In two batches, sear the beef, browning on all sides, about 6 minutes per batch. Transfer the beef to a plate.
Using an immersion blender, blend the chile and tomato mixture until smooth.
Add the blended sauce to the Dutch oven, scraping the pan to release any bits stuck to the bottom. Add the meat back to the pot with any juices from the plate. Add the apple cider vinegar, coriander, cumin, dried oregano, cinnamon sticks and bay leaves. Bring to a simmer, cover and transfer to the oven. Cook until the beef is very tender, about 3 hours.
Remove the beef to a bowl and shred with 2 forks. Set aside.
For the tacos: To build the tacos, grab a tortilla and dip it in the warm braising liquid. Lay the tortilla on a griddle over medium heat, top with a couple of tablespoons of the mozzarella and 2 to 3 heaping tablespoons of the beef. Add a little onion and cilantro, then fold in half to close. Repeat with 3 more tortillas.
Cook the tacos on the griddle until crisp, 30 to 45 seconds per side. Remove to a platter and squeeze over the lime juice. Repeat with the remaining tortillas and serve with the braising liquid on the side for dipping.