For garnish: chopped radishes, chopped cilantro, finely diced white onion, tortilla chips, shredded romaine lettuce, oregano, lime wedges
Place the pork shoulder in a large stock pot or dutch oven. If your pork shoulder has a bone, add that too. Add the onion, 5 of the smashed garlic cloves, and 2 teaspoons salt. Add water until it reaches 1 inch above the pork. Bring to a boil, then reduce to a simmer. Simmer on low until the meat is tender, about 2–3 hours. Remove the onion, garlic cloves, and bone (if you added it). Add the hominy, reserving 1 cup for later use. Cook for 30 minutes more.
Meanwhile, place the peppers in a medium bowl. Pour 5 cups of boiling water over the top and cover. Let sit for 20 minutes.
Drain the peppers. Remove the stems and cut the peppers open. Remove the seeds and membranes.
Place peppers, remaining 5 cloves of garlic, 2 teaspoons oregano, 2 teaspoons cumin, and 3 teaspoons salt in a blender. Pour 1 cup water over the top. Blend on high until almost smooth.
Add the remaining 1 cup hominy and blend until completely smooth.
Pour the pepper mixture into the pot with the pork. Stir to combine. Simmer for 30 minutes.
Serve in big bowls with chopped radishes, chopped cilantro, finely diced white onion, tortilla chips, shredded romaine lettuce, and lime wedges on the side.