Roast poblano peppers under the broiler or over an open flame until blackened and blistered. Transfer to a plastic bag or covered bowl to steam. Once cooled, peel off the skins, remove seeds, and finely chop.

In a large pot, melt butter and olive oil over medium heat. Add diced onion and sauté until soft, about 3 minutes. Add garlic and cumin; cook for 20 seconds until fragrant. Stir in chopped roasted poblanos.

Pour in chicken broth, salt, and pepper. Bring to a boil, then reduce heat to a simmer.

Add chicken and cream cheese, stirring until melted and smooth. Slowly whisk in 2 cups of cheddar cheese until fully incorporated.

Divide soup into oven-safe bowls. Top each with remaining 1/4 cup cheddar and pepper jack cheese. Broil until cheese is bubbly and golden.

Garnish with chopped cilantro and serve warm.