Pat your pork dry, season generously with kosher salt and cracked pepper, and preheat a heavy enamel stock pot over medium-high heat. Drizzle in olive oil. Sear your meat, then put aside.
Drop your diced onion into the pan and sauté in grease left from your pork shoulder. Once your onions are translucent, splash some chicken broth in and gentle scrap at the bottom to deglaze the pot.
Add your pork back to the pot, alongside cumin, oregano, and the remainder of your chicken broth. Reduce your heat to low.
Cut the stem off of your chilies and place them in a bowl alongside garlic. Pour about 3/4 cup of boiling water of your chilies, and let them sit for about 30 minutes.
When chilies are soft (after about 30 minutes), carefully use your immersion blender and pulse until your chilies, garlic, and water are smooth. Pour the mixture through a mesh sieve into your soup pot. Squeeze in the juice of one lime. Stir, and cover. Allow to simmer for 4-5 hours, stirring occasionally.
Dish your soup up and top with shredded cabbage, fresh lime wedges, cilantro, and fresh radishes.