Turn on the broiler. Place tomatillos and poblano on a baking sheet and pop under the broiler. Keep an eye on them, flipping them once when the first side is starting to char. Flip and let the other side char for about 4 minutes. Remove and let cool. Once cool, remove the stem and seeds from poblano and dice.
In a Dutch oven, heat vegetable oil over medium heat. Cover chicken breasts in salt, pepper and green chili powder (if you have it). Add chicken to oil and let first side cook for 5 minutes. Flip and cook the other side for 5 minutes. You just want to get a crust on the chicken, not cook it through. Remove to a plate and set aside.
Add more oil to the Dutch oven and add the onions and can of green chilies. Add salt and pepper, and let cook for 5 minutes. Add in the garlic and let cook for another minute.
Add the diced poblano and tomatillos to the pot, adding a dash of salt and pepper. Let everything come together for 3 minutes. Now add the chicken stock and cumin, and a dash more salt. Bring to a simmer. Once simmering, turn off heat.
Add lime juice and cilantro to the pot. Using an immersion blender, blend until smooth. Taste for seasoning and add salt/pepper as needed.
Add chicken and all the juices collected on the plate into the pot. Turn heat back on to low, cover and let cook for 1 hour. After 1 hour, shred the chicken into the juices, and let simmer for anther 30 minutes. If it won't shred after an hour, just let it go for a little longer.
Serve the shredded chicken, burrito bowl style, on tacos or burritos, or just eat it by itself. It's so tasty!