Preheat oven to 375 and line a baking sheet with parchment.
Add the poblanos and tomatillos. Drizzle with olive oil and toss to coat. Roast until the poblanos are tender and the tomatillos have softened, 15 minutes.
Meanwhile, heat a large pot over medium high heat. Add the chicken broth and bring to a simmer.
Meanwhile, in a high-speed blender, add the white onion, garlic, kosher salt and 1-2 ladles of the broth (or approx. 1 cup). Blend until smooth then add into the soup pot. (don't put up the blender-- you'll need it a few times).
In the blender, add the green leaf lettuce, cilantro and 2-3 ladles of the broth mixture (about 2 cups). Blend to combine then add to the soup pot.
Once the cook time on the poblanos and tomatillos is complete, add to the blender with 1-2 ladles of the broth (you may need to do this in 2 batches depending on the size of your blender). Blend until smooth and add to the soup pot.
At this point in the cooking process there may be some foam forming at the top of the soup. Using a spoon, skim the foam off the top of the soup and discard.
Add the chicken breasts into the soup and cook, simmering over medium-low heat, until the chicken is cooked through and tender, about 1 hour. Once the chicken is cooked through, using tongs remove the chicken from the pot and onto a cutting board. Using two forks, shred the chicken or dice up the chicken (I prefer it diced).
Add hominy and stir to combine. Simmer until the hominy is just tender, 15 to 20 minutes. When the hominy is just tender, add the chicken back into the soup. Stir to combine. Keep over low heat until ready to serve.
When ready to serve, ladle soup into bowls and garnish with sliced radishes, iceberg, lime, avocado and cilantro. Serve and enjoy!