Prepare a charcoal grill. Open the bottom vent completely. Light a large chimney starter full of briquettes. When the top coals are partially covered with ash, pour evenly over grill. Add the grate, cover and open vent completely. Heat until hot, about 5 minutes.

Alternately, heat a gas grill to high heat and cover.

Clean and oil the cooking grate. Add the chiles and onion and cook until lightly charred and tender, 2-4 minutes per side. Transfer to a bowl, cover and allow to sit for 5 minutes. Remove skins from chiles and discard; separate the onion rounds into rings.

Transfer the chiles and onion to the bowl of a food processor. Add garlic and pulse 10 times or until coarsely chopped. Remove all but ¼ cup of the chile mixture to a bowl and season with salt and pepper to taste. Process the remaining mixture until smooth.

Combine the beef, chile puree, ½ teaspoon of salt and ¼ teaspoon of pepper in a large bowl. Gently combine with your hands until well incorporated. Divide into 4 portions. Working with one portion at a time, lightly form into a ball. Gently flatten into a ¾-inch-thick patty. Press center of each patty down with your fingertips to create a ¼-inch-deep depression.

Place burgers on the grill and cook, covered, until browned on the first side, 3-5 minutes. Flip, then top each with the chile mixture and a slice of cheese. Continue to grill, covered, until the cheese has melted and burgers are cooked to your desired doneness, 3-5 minutes longer. Split buns and grill, if desired. Serve burgers immediately.