Line a baking sheet with aluminum foil for easy cleanup if desired, and place the poblano peppers on top. Set your oven rack directly underneath the broiler and turn the broiler on high.

Broil poblanos in the oven for 5 minutes or until the skin is blackened and blistered. Carefully flip them over and broil for another 5 minutes or until the skin is blackened and blistered on all sides.

Remove them from the oven, loosely cover the baking sheet with aluminum foil or plastic wrap to keep in some of the heat to help them steam, and let them sit for 5 minutes.

Transfer poblanos to a cutting board, slice open, and discard the stems and seeds. Dice into small chunks and set aside.

Add the butter and oil to a large pot or Dutch oven over medium heat. When the butter has melted, add the onions and saute for 5-7 minutes until soft and translucent.

Add the garlic and cumin and cook for 30-60 seconds until fragrant.

Add the flour and stir until the onions are evenly coated. Reduce the heat to low and cook for an additional 2 minutes, stirring occasionally.

Stir in the chicken broth and heavy cream. Continue stirring until the soup begins to thicken, then slowly add the Monterey jack and sharp cheddar cheese, stirring constantly until the cheese is fully melted and incorporated.

Raise the heat to medium-high and add in the roasted and chopped poblano peppers. Bring the soup to a simmer, reduce heat to low, cover, and cook for 15 minutes.

Serve in bowls with more shredded cheese or toasted panko breadcrumbs if desired.