Preheat oven to 400ºF.
To an oven-safe stockpot, add the onion, poblano and jalapeño peppers, shallot, and garlic. Drizzle with olive oil, add the salt, and mix. Place the pot into the oven and bake for 25 minutes, stirring halfway.
Remove the stockpot from the oven and add the green chiles, salsa verde, chicken broth, and chicken breast. Mix well.
Place the pot on the stovetop over medium/high heat. Bring the mixture to a boil. Once boiling, cover, and reduce heat to low. Let simmer for 25-30 minutes, or until chicken is fully cooked.
Once the chicken is cooked through, transfer the chicken to a plate or cutting board and use 2 forks to shred it. Set aside.
Use a handheld immersion blender to blend the contents of the pot just until there are small chunks.
Add the shredded chicken back into the pot along with the beans and hominy.
Stir and let simmer for an additional 10 minutes.