In a large soup pot, add chicken pieces, quartered onion, garlic, bay leaves, salt and pepper. Cover everything with water and allow to simmer until chicken is cooked through and tender. As water begins to evaporate, add hot water to keep the broth at the same amount and use a ladel to skim any foam at the top of the stock.
While stock is cooking, make the chile sauce. Cut open your dried chile's and remove all the seeds and stems. Soak the chiles in warm water for at least 30 minutes to soften.
When chiles are soft, add them to your blender (or food processor) with the rest of the chile ingredients; the onion, garlic cloves, few sprigs of fresh oregano, drizzle of olive oil and about 1/2 cup of the chile water (from soaking). You may need more depending on consistency. Blend everything together until it is all pureed.
When stock is done and chicken is cooked through, remove chicken to cutting board and shred or cut into small pieces. Discard stocks onion, garlic and bay. Then add the pureed chile sauce to chicken stock and stir together. Then add the hominy, black beans and shredded chicken. Taste for seasoning and continue cooking for another 30 minutes to blend flavors together and soften beans.
While soup is simmering, bring another small pot of water to a boil for the matzo balls.
In a bowl add matzo meal, masa, eggs, sparkling water, grapeseed oil, paprika, lime zest and salt and pepper. Mix everything together and place in fridge to set for about 20 minutes.
When matzo mixture is ready, it will not be as "lose" as when you first mixed, but will be a little firmer. With wet hands, roll the matzo balls lightly between your palms, about the size of a golf ball.
Add matzo balls to simmering water, a few at a time and cook until they float to the top. Once done, add them to the finished soup or store separately.
Serve pozole with matzo balls and garnish with cabbage, radish and lime.
For leftovers, store the matzo balls separate from the soup to last longer.