Place the chicken thighs, tomatillos, onion, garlic, poblano peppers, chicken broth, cumin, and oregano in a large pot. Season with salt and pepper. Bring to a boil over medium-high heat, then reduce to a simmer.
Cook uncovered for 20 minutes until the chicken is cooked through and the vegetables are soft. Remove the chicken to a plate and shred it with two forks.
Add the cilantro to the pot. Use an immersion blender to puree the broth and vegetables until mostly smooth, leaving some texture if desired.
Stir the shredded chicken and hominy back into the pot. Simmer for 10 more minutes to heat through and let the flavors blend. Adjust salt and pepper as needed.
Ladle into bowls and serve with lime wedges, radishes, avocado, and cabbage on the side.