Place both kinds of dried chiles into a small container. Pour the just boiled water over them. Try to get the peppers submerged under the water as best as you can. Let the peppers soak for about fifteen minutes, stirring them occasionally. You can prep your other ingredients in the meantime.
Transfer the chiles and their soaking water to a blender. Blend until smooth. Set this aside for now.
Coat the bottom of a large pot with the oil and place it over medium heat. Once the oil is hot, add the onion. Sweat the onion for about five minutes, stirring occasionally, until it becomes soft and translucent.
Stir in the minced garlic, ground cumin, oregano, ancho chile pepper, and smoked paprika. Sauté the mixture for about a minute, stirring constantly.
Stir in the broth, hominy, black beans, pinto beans, tomato sauce, bay leaves, and blended chile peppers. Raise the heat and bring the mixture to a boil. Lower the heat and let the mixture simmer for 30 minutes, stirring occasionally.
Remove the pot from heat. Remove the bay leaves and season the stew with salt and pepper to taste.
Ladle into bowls, then top each with shredded cabbage, sliced avocado, radishes, and chopped cilantro. Serve with lime wedges on the side.