Trim away the layer of fat from the short ribs with a sharp knife. Reserve the fat.

Trim away all the fat and anything that is not red, but it is not a big deal if some traces remain behind.

Toast the dried chiles in a frying pan without fat over medium high heat. The chiles should become soft and fragrant, but not burnt.

Remove the stems and seeds from the chiles, place them in a bowl, and cover with cold water. Allow them to soak for at least 10 minutes.

Roast 4 plum tomatoes, 1 quartered onion, and 8 unpeeled garlic cloves under the broiler.

Toast cloves, allspice, cumin seed, and black peppercorns in the oven for 5 to 10 minutes.

Grind the toasted spices with a spice grinder or mortar and pestle.

Puree the soaked chiles with the roasted tomatoes, garlic, onion, ground spices, and salt using a blender.

Separate the short ribs into individual ribs, vacuum seal them with the mole, and cook sous vide for 48 hours at 68C/155F.

Prepare the salsa by roasting tomatoes and garlic, sautéing arbol chiles, and blending all ingredients.

Sieve the gravy and discard the solids. Keep the gravy warm.

Melt beef fat in a frying pan and brown the beef chunks on all sides.

Pull the beef into shreds and assemble the quesadillas with cheese and beef.

Fry the quesadillas until nicely browned on both sides.

Serve the quesadillas with cilantro, minced onion, salsa, and warm birria gravy.