How to Make It
Heat oil in a large pot over medium-high. Season chicken with 1 teaspoon salt and ½ teaspoon pepper. Working in batches, add chicken and cook until browned, 5 to 6 minutes per side. Transfer to a plate.
Reduce heat to medium. Add onions, fresh chile, oregano, and remaining 1½ teaspoons each salt and pepper. Cook, stirring occasionally, until onions are soft, 6 to 8 minutes.
Add ½ cup broth; scrape brown bits from pot with a wooden spoon. Return chicken to pot along with remaining 7½ cups broth. Bring to a boil, cover, and reduce heat. Simmer until chicken is cooked through, 20 to 30 minutes; transfer chicken to a plate. When cool enough to handle, shred chicken into bite-size pieces with 2 forks. Return to soup (discard skin and bones).
Meanwhile, puree canned chiles and 2 cups cilantro in a blender until smooth.
Stir chile puree and hominy into soup. Simmer until warmed, about 5 minutes. Serve topped with tortilla chips, cilantro, and sliced fresh chiles; place lime wedges on the side.
How to Freeze: Cool soup completely. Pour into gallon-size freezer bags, filling halfway and squeezing out air. Seal and stack the bags in the freezer for up to 3 months. Thaw in the refrigerator overnight or in a bowl of cold water for 10 minutes. Reheat in a pot over medium until warmed through, 8 to 10 minutes.