Season the pheasant legs and thighs with 2 teaspoons of the kosher salt.
Set instant pot or other pressure cookerto sauté setting. Add oil. Once the oil is hot, add the pheasant, and cook for about 3 minutes on each side, until meat is nicely browned.
Add 2 cups of the chicken broth with ½ teaspoon each black pepper, cumin, and Mexican oregano to the pressure cooker and mix well. Now secure lid on the pressure cooker and pressure cook for 30 minutes. Then allow the pressure to release naturally for 15 minutes.
Use this time to slice and chop your vegetables.
Transfer pheasant meat to a large plate and allow the pheasant meat and bones to rest and cool for at least 15 minutes. Reserve all the liquid and transfer it to a bowl to use later.
Meanwhile, add one cup of water to a blender. Then add tomatillos, yellow onion, garlic, one of the jalapeños, and one bunch of the cilantro. Blend until smooth, using a spatula to scrape down sides if necessary.
In a large Dutch oven, heat the remaining 1 Tablespoon olive oil and add tomatillo puree and cook over medium heat, stirring occasionally, until the sauce turns a deep green, about 5 minutes.
Add the reserved liquid used to cook the pheasant, the remaining 4 cups of chicken broth, the hominy with liquid, and another ½ teaspoon each of kosher salt, black pepper, cumin, Mexican oregano, and chili powder, and stir well to combine. Cover and bring the stew to a simmer over medium-low heat for about 10 minutes.
The pheasant meat should be cool at this point, making it easy to pull off the bones using your fingers or tongs. Discard the bones and tendons. Stir the shredded pheasant meat into the stew with a squeeze of fresh lime juice from one of the lime quarters.
Taste the green pozole and adjust the seasonings if necessary. Ladle the stew into individual bowls and serve immediately with the remaining lime wedges, radish, red onion, and cilantro on the side so people can add whatever toppings they want.