In a medium saucepan, bring the chicken broth to a boil. Add the chicken breast, and bring the broth to a boil again. Cover the pot, turn off the heat, and let stand until ready to use.
Trim, peel, and cut the red onion into large chunks. Trim, peel, and cut the garlic into chunks. Quarter the tomato. Trim and dice the green pepper. Tie the parsley sprigs together with kitchen twine.
In a blender, combine the onion, garlic, and tomato with the ancho chile powder, salt, cumin, oregano, and black pepper and blend to make a paste. (Add a little water if the paste needs thinning.)
In a large, heavy pot, sauté the diced green pepper in the corn oil over low heat, stirring, until soft. Add the spice paste and cook until somewhat dry, about 5 minutes. (It may splatter some.)
Remove the chicken breast from the broth. Add the broth to the paste mixture, stirring well to combine. Add the parsley and simmer for 30 minutes.
On a cutting board, shred the chicken with 2 forks into thin strands.
Drain and rinse the hominy well in a colander under cold water. Add the hominy and chicken to the soup and bring it to a boil. Correct the seasonings and remove the parsley.
Sprinkle the cilantro over the soup. Serve immediately in bowls with a dollop of crème fraîche and corn chips on the side.