Heat a Dutch oven or pot over medium heat, add 2 tbsp of olive oil & heat through. Season pork with salt & pepper. Working in batches, sear pork on all sides (~8 min). Return pork to pot & add 4 cups of water.
Lower the heat to medium-low, cover & simmer until meat is tender (~1 hour).
Bring a large pot of water to a boil over medium-high heat, add tomatillos, jalapeños & poblano pepper & simmer (~15 min). Remove from heat, drain & set aside.
Heat a large sauté pan over medium heat, add remaining olive oil & heat through. Add onions & garlic & sauté, stirring occasionally, until onion is soft & translucent (~8 min).
Remove from heat & add to a blender with tomatillo mixture cumin & oregano. Blend the mixture until pureed & smooth. Set aside.
Remove pork from pot & set aside. Skim fat from the pot & reserve cooking liquid. When pork is cooled, shred it & discard any excess fat. Pork will become more tender as soup cooks!
Return shredded pork to pot with cooking liquid, tomatillo puree, chicken stock & hominy. Continue to simmer until pork is falling apart & hominy increases in size (additional ~30 min).
Ladle pozole into soup bowls & top with cilantro, cotija cheese, jalapeño, radish & avocado as desired. Serve immediately.