Put the garlic onto a piece of aluminum foil and drizzle with a little olive oil. Close up the foil and roast until the garlic is soft, about 45 minutes.
In a large skillet, heat a thin film of oil over moderate heat. Add the chiles and brown on all sides. Remove the chiles and set the skillet aside.
Generously season the meat with salt and pepper. Reheat the same skillet over medium heat and cook the meat until well browned on all sides, about 15 minutes. Do this in batches to avoid crowding the pan, adding additional oil as needed. Remove the meat with a slotted spoon and set aside. Pour off the excess fat from the pan. Deglaze the pan with about 1/2 cup water, scraping the browned bits off the bottom.
In a large pot, bring 4 quarts of water to a simmer and add the liquid from the skillet. Add the garlic, chiles, posole, oregano, and pepper to taste. Simmer 1 hour.
Add the browned meat. Simmer gently, partially covered, about 2 hours, or until the meat is very tender. Taste and adjust the seasoning.
Serve the posole in soup bowls with the garnishes in individual bowls on the side.
Felicitas' Hot Salsa
In a large skillet lined with aluminum foil, toast the chiles over moderate heat for about 5 minutes, until they almost smoke, stirring often.
Bring 2 cups of salted water to a boil. Add the toasted chiles, garlic, and oregano. Simmer about 10 minutes, or until the chiles are soft.
Pour into a blender. Blend on low speed for several seconds, then increase to high speed for about 1 minute or until smooth. Add water to thin out sauce if you prefer.