Heat a large heavy bottomed Dutch oven or pot over medium heat, add 2 tablespoons of the olive oil and heat through. Season the pork with salt and pepper. Working in batches, sear the pork on all sides, about 8 minutes. Return the pork to the pot and add 4 cups of water.
Lower the heat to medium-low, cover and simmer until the meat is tender, about 1 hour.
Bring a large pot of water to a boil over medium-high heat, add the tomatillos, jalapeños and poblano pepper and simmer for 15 minutes. Remove from heat, drain and set aside.
Heat a large sauté pan over medium heat, add the remaining olive oil and heat through. Add the onions and garlic and sauté, stirring occasionally, until the onion is soft and translucent, about 8 minutes.
Remove from heat and add to a blender with the tomatillo mixture cumin and oregano. Blend the mixture until pureed and smooth. Set aside.
Remove the pork from the pot and set aside. Skim the fat from the pot and reserve the cooking liquid. When the pork is cooled, shred it and discard any excess fat. (It's ok if the pork is still slightly tough, it will continue to become tender as the soup cooks).
Return the shredded pork to the pot with the cooking liquid with the tomatillo puree, chicken stock and hominy. Continue to simmer until pork is falling apart and hominy increases in size, about an additional 30 minutes.