Pour 10 cups of water into a 6-quart stockpot and place over high heat. Add the chicken breasts, quartered onion, garlic cloves, and 1 teaspoon of salt. Once boiling, reduce the heat to medium-low and simmer gently for about 20 minutes, or until the chicken is fully cooked and tender.
Meanwhile, place the dried chiles in a bowl and cover with hot water. Let them soak for 10 to 15 minutes, or until they are soft and pliable. Discard the soaking water once softened.
Remove the chicken from the pot and allow it to cool slightly before shredding into bite-sized pieces. Set aside. Strain the broth if desired and reserve.
In a blender, combine the soaked chiles, garlic, onion from the broth, and 1 cup of the reserved chicken broth. Add the remaining teaspoon of salt and blend until smooth, forming a rich red sauce.
Return the shredded chicken to the pot along with the chile mixture and the hominy. Pour in the reserved chicken broth and bring the mixture to a gentle boil. Reduce heat and let it simmer uncovered for another 20 minutes, allowing the flavors to meld and the broth to deepen in color and taste. Adjust seasoning if needed.
Ladle the hot pozole into individual bowls. Serve with an array of garnishes—each guest can customize their bowl with fresh cabbage, radishes, lime, onion, and a sprinkle of oregano. Pair with crispy tostadas on the side.