Season the chicken breast and thighs with salt on each side. In a large 6-quart Dutch Oven or large stockpot, over medium heat, add 1 tbsp cooking oil. Pan-sear the chicken breast and thighs, cooking it on both sides for 1-2 minutes.
Add the chicken broth, water, garlic, onions, and bay leaves to the pot.
Bring the liquid to a boil. Then, reduce the heat to medium-low and simmer covered for 1 hour.
Remove the bay leaves. Remove the chicken from the pot and place it on a plate or cutting board. Shred the chicken and discard the bones. Add the shredded chicken back into the broth.
Reserve 1.25 cups of broth and set aside.
Rinse the hominy, then add it to the pot. Allow the mixture to continue to simmer as you prepare the sauce.