Preheat and roast the poblanos. Place peppers on a baking sheet and broil on high or roast at 450°F. Turn every few minutes until skins are evenly blackened and blistered, about 8 to 12 minutes.

Transfer hot peppers to a bowl and cover for 10 minutes. Slip off skins, remove stems and seeds, and roughly chop. Wear gloves if you are sensitive to pepper oils.

In a large pot, melt butter over medium heat. Add chopped onion and cook until translucent, 4 to 5 minutes. Stir in minced garlic and cumin and cook for 30 seconds until fragrant.

Add chopped chicken, chopped poblanos, and chicken bone broth to the pot. Season with salt and pepper. Gently simmer until the chicken is cooked through, about 10 to 12 minutes.

Remove about 3 cups of the soup into a blender. Add cream cheese and blend until completely smooth. Return the blended base to the pot and stir in shredded cheddar until melted. Keep the soup at a gentle simmer; avoid boiling after adding cheese.

Ladle soup into oven-safe bowls. Top each bowl with a slice of cheddar or pepper jack. Broil until the cheese is melted and bubbly, 1 to 2 minutes. Watch closely to prevent burning.