Preheat and roast the poblanos. Arrange poblanos on a baking sheet and broil on high or roast at 450°F until the skins are evenly blackened and blistered, about 8–12 minutes. Turn as needed for even charring.

Steam and peel. Transfer the hot peppers to a bowl and cover for 10 minutes to steam. Peel off skins, discard stems and seeds, and roughly chop the flesh.

Cook aromatics. In a large pot, melt the butter over medium heat. Add the chopped onion and cook until translucent, about 4–5 minutes. Stir in the garlic and cumin and cook 30–45 seconds until fragrant.

Simmer the chicken and peppers. Add the chopped chicken, chopped poblanos, and chicken bone broth. Season with salt and pepper. Bring to a simmer and cook until the chicken is cooked through, about 10–12 minutes.

Make the creamy base. Carefully remove about 3 cups of the hot soup and transfer to a blender. Add the room-temperature cream cheese and blend until smooth. Return the blended mixture to the pot and stir in the shredded cheddar until melted and combined. Taste and adjust seasoning.

Broil the cheese topping. Ladle the soup into oven-safe bowls. Top each bowl with slices of cheddar or pepper jack and broil on high until the cheese is melted and bubbly, watching closely so it doesn’t burn. Serve hot.