FoodNetwork.com and Saveur.com each have a recipe for Mexican birria. Saveur adapted its recipe from an authentic dish that takes over four hours to make, but Food Network's recipe only takes two hours.
To make Food Network's recipe, roast four guajillo chiles and four ancho chiles, chill them in cold water for 20 minutes, and puree the chiles with 1 cup warm water in a blender. In a deep Dutch oven, add 1 pound of cubed top round, 1 pound of cubed baby back ribs, 1 cup of hot water, 1 quartered onion, and 6 garlic cloves. Bring to a boil and cook for 1 hour. Add the chile puree, 2 fresh bay leaves, 6 sprigs thyme, 1 tablespoon ground cumin, 1 tablespoon Mexican oregano, salt and pepper. Simmer for 30 minutes. In a blender, puree 1/2 cup of red wine vinegar, 2 garlic cloves and 1 teaspoon of chili powder into a sauce. Garnish the stew with diced onion, lime slices and chopped cilantro and serve corn tortillas and the sauce on the side.
Saveur's recipe calls for 5 pounds of young goat or 3 pounds of lamb roast, rubbed in salt and roasted on a steaming rack in a large sealed pot with 4 cloves of crushed garlic, 1 finely chopped white onion, and 4 cups of water at 325 degrees for 4 hours. A pasilla chile is halved, toasted in a skillet and soaked in 1 cup of hot water for 30 minutes before pureed in a blender. Blend 1/2 teaspoon ground cumin, 2 teaspoons marjoram, black pepper, 1/2 teaspoon ground ginger, 2 teaspoons sesame seeds, 4 whole cloves, 4 teaspoons of sugar, 6 tablespoons of vinegar and 3 tablespoons of water into the chile puree. The meat is glazed with 3/4 cup of the chile sauce at 350 degrees for about 20 minutes, then shredded. A birria sauce is made by simmering, uncovered, 5 pounds of cooked, pureed tomatoes, the remaining chile sauce, 2 bay leaves, 1 small cinnamon stick and the water used to steam the meat until the liquid is reduced by one-third. The meat is served with the sauce and garnished with diced onion, lime slices and chopped cilantro.