In a large pot over medium-high heat, add the olive oil and sear the chicken for 3 minutes each side.
Once the chicken has been seared, add 2 quarts of water and boil for about 30 minutes until the chicken is fully cooked.
Remove chicken and set aside to cool, but leave the water in the pot.
Using a slotted spoon, skim the top of the water of impurities. Add chicken broth, hominy, onion, salsa, garlic, salt, pepper, oregano and bring to a boil. Simmer over medium-low heat for 30 minutes.
While the vegetables are simmering, shred the chicken.
Add chicken back into the pot and boil for another 30 minutes.
While the chicken is simmering with the rest of the ingredients, chop the cabbage, radish, avocado, and lime.