Cook the Chicken: In a large pot, combine the chicken, chicken broth, quartered onion, minced garlic, and dried chiles. Bring to a boil, then reduce heat and simmer until the chicken is cooked through (about 20-25 minutes).
Shred the Chicken: Remove the chicken from the pot and shred it using two forks. Discard the onion and chiles.
Add Hominy and Spices: Return the shredded chicken to the pot, add the drained hominy, cumin, oregano, salt, and pepper. Simmer for an additional 15 minutes to allow the flavors to meld.
Serve: Ladle the pozole into bowls and top with shredded cabbage, diced radishes, cilantro, and avocado. Serve with lime wedges on the side.