Start by gathering all the necessary ingredients for the recipe.
In a deep skillet, heat 2 tablespoons of vegetable oil over medium-high heat. Add the chicken breasts, cooking them until they are no longer pink and the juices run clear, approximately 20 minutes. Once cooked, remove the chicken from the skillet, drain, and allow it to cool. Once cooled, shred the chicken using a fork.
Using the same skillet, heat the remaining 2 tablespoons of vegetable oil over medium-high heat. Add the chopped onion and minced garlic, cooking and stirring until both are soft and translucent, which should take about 5 minutes.
Return the shredded chicken to the skillet. Stir in the chicken broth, water, chili powder, salt, and oregano. Lower the heat, cover the skillet, and let it simmer for around 90 minutes.
Afterward, add the hominy to the pot and cook until it becomes tender, which will take about 15 more minutes. Taste the pozole to adjust the seasonings, adding additional salt and chili powder if desired. Serve the pozole in soup bowls alongside a tostada shell, garnishing as preferred.