Heat the olive oil in a large pot over medium heat until the oil begins to smoke. Add the onions, garlic, chorizo and chili powder and cook until the onions are translucent and the pork is cooked, 4 to 5 minutes.
Add the chicken stock, crushed tomatoes and hominy and bring to a simmer.
Cut the chicken breasts into very thin slices. Place in a bowl with a pinch of salt and the lime juice. Mix together and let sit until the broth comes up to a simmer. Add the chicken to the soup and continue to simmer until cooked through, about 5 minutes. Season with salt.
Serve the soup in bowls topped with the shredded cabbage, shaved radish and jalapeno, sliced avocado, fresh lime wedges and cilantro.