Roast the chiles under the broiler until blackened on both sides, about 10-12 minutes, turning once. Once blackened, place the chiles in a paper sack or plastic food-storage bag, close it tight and let the chiles steam for 20 minutes. After the chiles have steamed, remove from the bag and rub off the skin. Remove the stems and seeds from both the chiles and chop.
To make the burgers, divide the ground beef into 8 equal portions and form into balls. Melt 1 tablespoon of the butter in a large cast-iron skillet over medium heat. Working in batches, place the buns, cut-side down, in the skillet and cook for 1 to 2 minutes, or until lightly toasted. Transfer the first batch to a plate, add the remaining 1 tablespoon butter to the skillet, and toast the remaining buns. Cover with aluminum foil to keep warm.
Using the same skillet, turn the heat to high. (If you don’t have a powerful hood, open a window for ventilation.) Place 2-4 meat balls in the skillet (depending on skillet size). Cook for 30 seconds, then smash down with a flat spatula to form a patty about 4 inches across. Season the patties with salt and pepper.
After 2 minutes, or when the patties have developed a crust on the bottom, flip them, top with a slice of cheese, and cook for 1 to 2 more minutes, or until the meat is cooked to your liking and the cheese has melted. Rest the patties on a plate and top with chopped green chiles. Repeat cooking process with the remaining meat balls.
Dress the buns with mayonnaise and/or mustard, if desired. Place lettuce, tomato, and/or onions on the bottom bun then top with the patties. Cover with the top bun and serve immediately.