Add the guajillo, ancho, and arbol chiles to a large bowl and cover with boiling water. Let them soak until softened, about 10 minutes. Remove the stems and seeds, then reserve the soaking liquid.
Add the softened chiles to a blender along with the vinegar, tomatoes, garlic, oregano, paprika, cumin, and 2 cups of the soaking liquid. Blend until smooth.
Heat the oven to 300ºF (150ºC). Season the beef cubes generously with salt and pepper. Heat oil in a large Dutch oven over medium heat, then sear the beef in batches until browned on all sides, about 8 minutes total.
Add the seared beef back to the pot and top with the chili mixture, onion, beef stock and 2 cups of water. Wrap the bay leaves, thyme, cinnamon stick and cloves in cheesecloth, tie to close and then add to the pot.
Bring to a boil, then cover and transfer to the oven. Braise until the beef is fork-tender, about 3 hours.
Remove the meat from the braising liquid to a cutting board. Let cool slightly, then chop the meat with a cleaver or large knife. Season the consommé broth and reserve for serving.
For assembling the grilled cheese:
Heat a large non-stick pan over medium-low heat. Spread the mayonnaise on one side of each of the bread. Add ½ tablespoon (8g) of the butter to the pan and allow to melt. Working in batches, place 2 slices of the bread, mayonnaise-side-down on the pan, and top each with ⅓ cup (65g) of the cheese. Pile ⅓ cup (75g) of the chopped birria meat onto one of the slices of bread.
Add an ice cube to the pan and cover, cooking until the cheese has melted, about 2 to 3 minutes. Close the sandwich and continue cooking until crispy and golden-brown on both sides, about an additional minute on each side.
Remove to a cutting board and cut in half. Repeat with the remaining sandwiches.
Serve the birria grilled cheese hot with a cup of the consommé for dipping, topped with some chopped onions and cilantro.
Notes
The recipe for birria will make more than you need for the 4 grilled sandwiches but since it takes a bit of time I like to make the full batch and save or freeze half for another use.