Begin by roasting 4-6 poblano peppers to achieve that signature smoky flavor. Position the peppers over an open flame or under your oven's broiler, turning occasionally for even charring, until the skin blisters and blackens, about 10-15 minutes. Once roasted, transfer them to a bowl and cover with plastic wrap to steam for 10 minutes, which makes peeling easier.

After the poblano peppers have cooled, carefully peel off the charred skin using your fingers or a paper towel. Be sure to remove the seeds and stems for a smoother texture. Chop the roasted peppers into small pieces, ready to add to your creamy Chile Relleno Soup and enhance its depth of flavor.

In a large saucepan, melt 2 tablespoons of butter over medium heat. Once the butter is melted and bubbly, add one finely chopped onion and sauté for approximately 3-4 minutes, or until it becomes translucent. Stir in 2 minced garlic cloves and 1 teaspoon of ground cumin, allowing the mixture to cook for an additional minute.

Pour in 4 cups of chicken bone broth, seasoning it lightly with salt and pepper to taste. Bring the mixture to a gentle boil, then reduce the heat to low, allowing the soup to simmer. Add 1 pound of boneless, skinless chicken thighs or breasts, ensuring they are fully submerged in the broth, and cook for about 15-20 minutes, or until the chicken is cooked through and tender.

Remove the cooked chicken from the pot and shred it using two forks. In a blender, combine 4 ounces of cream cheese, 1 cup of hot broth, and 1 cup of shredded cheddar cheese. Blend until smooth and creamy. Return this mixture back into the soup pot along with the shredded chicken, stirring well to incorporate the rich flavors of your Chile Relleno Soup.

Ladle the flavorful Chile Relleno Soup into your broiler-safe bowls. Top each bowl with slices of cheddar or pepper jack cheese. Place the bowls under the broiler for 2-3 minutes, or until the cheese is bubbly and golden.