Place chiles on a paper towel-lined microwave-safe plate. Microwave on HIGH for 60-90 seconds, or until puffed and fragrant. Once the chiles are cool enough to handle, remove the seeds and stems.
Pour 2 cups of stock into a medium microwave-safe bowl. Add the chiles, cover the bowl with plastic wrap, and microwave for 2 minutes or until bubbling.
Let the mixture sit for 15 minutes so the chiles can soften.
Use paper towels to pat the pork dry and season all sides with salt and pepper. Add 2 tablespoons of the vegetable oil to a heavy-bottomed pot and heat over medium-high heat until simmering and small wisps of smoke are coming off of the oil.
Cook until the pork is brown on all sides, about 10 minutes. Transfer the cooked pork to a plate.
Add the hominem to the pot and cook, constantly stirring, until the hominy is fragrant and it begins to darken in color, about 3-4 minutes. Move hominy to a bowl.
Add the remaining 1 Tablespoon of oil to the pot and heat over medium heat until the oil is simmering. Add the onion and cook until softened and beginning to brown around the edges, about 5 minutes. Add the garlic and stir, and cook until fragrant, about 30-60 seconds. Use an immersion blender or blender to puree the onion mixture and the chile mixture. Add the onion-chile mixture back to the pot.
To the pot, add in the remaining 6 cups of chicken stock and stir to combine.
Add the cooked pork, oregano, 1/2 teaspoon of salt, and 1/2 teaspoon of pepper, and bring it to a boil over medium-high heat. Reduce the heat to low and simmer, covered, until the meat is nice and tender, 75-90 minutes.
Remove the pork from the pot and place it on a plate. Add the hominy to the pot and simmer, covered for 30 minutes. Use a spoon to skim the fat from the top of the soup. Shred the pork, discarding the fat, and return the pork to the pot and cook until the pork is heated through about 1 minute.
Turn the heat off and squeeze in the lime juice, and season the Posole with salt and pepper to taste.
Ladle the soup into bowls and garnish with shredded cabbage, diced avocado, sliced radishes, chopped white onion, and lime wedges (all optional).