Step 1Remove the stems and seeds from the pasilla and guajillo chiles and discard. Place the chiles in a saucepan and cover them with 4 cups of water. Bring to a boil, then remove the pan from the heat and set it aside for the chiles to soften.
Step 2Dice 1 onion and set it aside. Cut the second onion in half and dice half of it. Cut the remaining half lengthwise into slices.
Step 3Working in batches, pour the chiles and their cooking liquid, 1 1/2 diced onions, the garlic, pepper and 1 tablespoon of oregano into a blender and puree until smooth.
Step 4Add the remaining 6 cups of water to the chile puree and strain through a fine mesh sieve, using a spoon to push all of the chile juice into a large stock pot. Throw away the solids.
Step 5Add the chicken pieces, hominy and onion slices to the pot and bring to a boil. Simmer until the chicken is tender and comes off the bone easily, about 1 hour. Add the remaining oregano and salt to the soup. Remove the chicken from the pot. Remove the meat from the bones, return it to the soup and heat through, discarding the bones. Skim the fat from the top of the soup before serving.
Step 6Garnish with the cabbage, radish slices, onion, lime wedges and avocado.