Roast green chile peppers until skin is charred and blistered. You can do this over the open flame on a gas stove, or by heating your broiler to high and placing chile within a few inches of the broiler (turning to get all sides charred).
Place green chile in a bowl and cover with plastic wrap. Let cool, then peel skin as much as possible. Cut out seeds and chop by hand or place in a food processor / blender to finely chop. Set aside.
In a large saucepan over medium heat, melt butter. Add onion and cook, stirring frequently, until translucent, about 5 minutes. Add garlic, cumin, and green chile and stir until fragrant, about 1 minute.
Stir in chicken broth and season to taste with salt and pepper. Bring to a boil and then reduce heat to a simmer. Add chicken.
Add cream cheese and 2 ½ cups of the cheddar cheese and whisk until smooth.
To serve, divide into 6 bowls and sprinkle each with ¼ cup shredded cheddar cheese. Sit briefly under the broiler until cheese is melted and browned (this step is optional but OH SO GOOD!).
