Prepare the chicken by heating the oil in a large Dutch oven over medium-high heat.
Season both sides of the chicken breasts well with sea salt and freshly cracked pepper, to taste.
Place the breasts skin side down in the HOT Dutch oven and cook undisturbed for 4 minutes, until the skin is golden brown.
Flip the breasts over and cook the other side for 2 minutes.
Remove from the Dutch oven and set aside on a plate to cool a bit.
Once the chicken has cooled, remove the skin and discard.
Prepare the chile peppers and onion while the chicken is cooking by putting half of the Anaheim, poblano, and onion in a food processor for 12 (1-second) pulses, until consistency of chunky salsa.
Transfer mixture to bowl. Repeat with the remaining Anaheim, poblano, and onion.
Combine with the first batch. Don't bother washing processor - you will use it again.
Once the chicken has been removed from the Dutch oven, carefully remove all but 1 tablespoon of grease from the Dutch oven, or adding additional vegetable oil if necessary.
Reduce the heat to medium. Add the minced jalapeno, chile-onion mixture, minced garlic, cumin, coriander, and 1/4 teaspoon of salt.
Cover and cook, stirring occasionally, until vegetables soften, about 10 minutes. Remove pot from the heat.
Transfer 1 cup of cooked vegetable mixture into the empty food processor.
Add 1 cup of drained & rinsed beans and 1 cup of chicken broth.
Pulse for 20-30 seconds, or until creamy. Pour this mixture back into the Dutch oven.
Finish preparing the chili by adding the remaining 2 cups of chicken broth along with the chicken breasts to the Dutch oven and bring to a boil over medium-high heat.
Reduce the heat to medium-low and simmer, covered, stirring occasionally, until the chicken registers 160 degrees in the thickest part of the breast (or 175 degrees if using thighs) on an instant read thermometer, about 20 minutes (up to 40), depending on the size of the breasts (or thighs).
Using tongs, transfer chicken to a large plate to cool.
Stir in the remaining beans and continue to simmer, uncovered, until beans are heated through and the chili has thickened slightly, about 10-15 minutes.
Shred the chicken once it has cooled and discard the bones.
Add the chicken (and any juices from the plate), lime juice, and cilantro; stir to combine.
Taste and season with sea salt and freshly cracked pepper, to taste.
Ladle into bowls and serve topped with sour cream, diced avocado, green onions, fresh cilantro, along with tortilla chips and lime wedges on the side. Enjoy.