Heat 2 tablespoons oil in a deep skillet over medium-high heat.
Add chicken breasts, and cook until no longer pink and juices run clear, about 20 minutes. Remove from skillet, drain, and cool. When completely cooled, shred chicken with a fork.
Heat remaining 2 tablespoons oil in the same skillet over medium-high heat.
Cook and stir onion and garlic until soft and transparent, about 5 minutes. Return shredded chicken to the skillet.
Stir in the chicken broth, water, oregano, salt, and chili powder.
Reduce heat to low, cover, and cook about 90 minutes. Stir in the hominy and cook until tender, about 15 minutes more. Add more salt and chili powder, if desired.
Garnish and serve in soup bowls with1 tostada shell per serving.