Place poblanos over an open flame or grill, turning until charred on all sides (5–8 minutes). Transfer to a bowl, cover with plastic wrap, and let steam for 20 minutes.

In a blender, combine bone broth, softened cream cheese, shredded Cheddar and Monterey Jack, and salsa verde. Blend until smooth and creamy.

Peel off the charred skins and remove the seeds. Pulse the peeled peppers in a food processor until finely chopped.

Melt butter in a skillet over medium heat. Add onion, garlic, and cumin. Cook for 3–5 minutes, stirring, until lightly browned and fragrant.

Stir in the chopped poblanos and cook for 2–3 minutes. Add the shredded chicken and cook for another 2 minutes.

Pour the blended cheese mixture into a stockpot over low heat. Stir in the poblano and chicken mixture. Simmer for 10–15 minutes, stirring often. Season with salt and pepper to taste.