Roast the poblano peppers until the skin is blackened and blistered. Place in a bowl and cover with plastic wrap for 10 minutes to steam. Peel the skin, remove the seeds and stems, and chop.

In a large pot or Dutch oven, melt the butter and olive oil over medium heat. Add the diced onion and cook until softened, about 5 minutes. Add the minced garlic and cook for another minute until fragrant.

Add the chopped roasted poblano peppers and chicken broth to the pot. Bring to a simmer and cook for 15 minutes to allow the flavors to meld.

Stir in the cooked chicken breast, cream cheese, and Monterey Jack cheese. Continue to cook until the cheese is melted and the soup is smooth and creamy.

Season with salt and pepper to taste. Serve hot.