Set your oven broiler to high. Place poblano peppers on a baking sheet and broil for 5-7 minutes per side, until the skins are blackened and blistered. Transfer to a bowl and cover tightly with plastic wrap. Let steam for 10 minutes.

Once cooled, peel the blackened skins off the peppers. Remove the stems and seeds, then roughly chop the peppers.

In a large pot or Dutch oven, heat olive oil over medium heat. Add the chopped onion and cook until soft, about 5-7 minutes. Stir in the minced garlic and cook for 1 more minute until fragrant.

Transfer the onion, garlic, and chopped poblano peppers to a blender. Add 1 cup of chicken broth and blend until completely smooth.

Pour the blended mixture back into the pot. Stir in the remaining 3 cups of chicken broth, cumin, oregano, salt, and pepper. Bring to a simmer.

Reduce heat to low. Whisk in the softened cream cheese until fully melted and smooth. Stir in the heavy cream and the shredded chicken. Heat gently for 2-3 minutes, ensuring it does not boil.

Taste the soup and adjust seasoning if necessary. Serve hot, garnished with your favorite toppings like fresh cilantro, tortilla strips, or sour cream.