In a small saucepan over medium heat, add the drained hominy and heat it. Keep the hominy warm.
In a big, heavy pot over high heat, add the pinto beans, 10 cups of the water, and the 1/2 peeled white onion.
Bring the water to a boil.
Reduce the heat to medium and partially cover the pot.
Allow the beans to simmer until softened, about 1 hour 30 minutes.
Add 1 tablespoon of the kosher salt to the beans and stir to combine.
Simmer the beans until they are soft enough to come apart in your fingers, about 15 minutes. The water should have thickened into a thick broth.
Using a slotted spoon, remove the onion half. Set the frijoles de olla aside.
In a large soup pot over high heat, add the chicken and enough of the extra water to cover it by about 2-3 inches.
Bring the water to a boil.
Reduce the heat to medium-low and skim off any foam that has risen to the top of the water.
Add the remaining 1/2 peeled white onion, 4 of the peeled garlic cloves, the quartered Anaheim chile, 10 sprigs of the cilantro, and 2 teaspoons of the kosher salt to the chicken mixture and stir to combine.
Cover the pot partially and allow the chicken mixture to simmer until the meat is falling away from the bones, about 1 hour.
Transfer the pot with the chicken mixture from the heat.
Transfer the chicken pieces to a large bowl and allow it to cool until they can be safely handled.
Strain the chicken cooking broth, keeping the broth and discarding the solids.
Reserve 1 cup of the cooking broth and add the remaining broth to the soup pot.
Remove the skin and the bones from the chicken.
Cut or tear the chicken meat into small pieces.
Add the chicken meat to the broth in the soup pot.
Adjust an oven rack to the highest position possible and preheat the broiler.
Line a baking sheet with foil.
Add the whole Anaheim chile, the halved quarter of the remaining white onion, and the remaining garlic cloves to the prepared baking sheet.
Broil the chile and vegetables, flipping them halfway through, until completely charred on the outside, about 10-12 minutes. The garlic will be done much earlier, about 5-6 minutes, and should be transferred from the oven at that time.
Transfer the charred whole Anaheim chile to a plastic bag and seal the bag.
Allow the chile to sweat for 5-10 minutes.
Remove and discard the chile skin, stem, and seeds.
Peel the charred garlic cloves.
Cut the roasted and peeled chile into pieces.
In a blender, add the roasted chile, charred onions, the roasted and peeled garlic cloves, 10 sprigs of the remaining cilantro, and the reserved cup of the chicken broth and cover the blender.
Purée the chile mixture until completely smooth and it becomes a sauce.
Add the sauce to the chicken and the broth in the soup pot.
Add 1/2 of the frijoles de olla and 1 cup of the broth to the chicken soup in the soup pot.
Add the heated hominy to the chicken soup in the soup pot.
Place the soup pot over medium heat and bring it to a simmer.
Partially cover the chicken soup until the flavors come together and the sauce rises to the surface in dark green puddles.
Serve garnished with the remaining chopped onions, the remaining cilantro leaves, the crushed chiletpin chiles, and the lime wedges.