Step 1: If using fresh chiles, roast them over an open flame, under a broiler, or on a grill until the skin is blistered and charred. Place them in a bowl, cover tightly with plastic wrap, and let steam for 10-15 minutes. Once cooled, peel off the skin, remove stems and seeds, then chop coarsely. (If using canned chiles, drain thoroughly).
Step 2: Divide the ground beef into 4 equal portions. Gently form each portion into a ¾-inch thick patty, slightly wider than your buns. Press a shallow indentation into the center of each patty to prevent bulging.
Step 3: Season both sides of the patties generously with salt and pepper.
Step 4: Heat a large skillet or grill over medium-high heat. Cook the patties for 4-5 minutes per side for medium-rare, or longer for desired doneness.
Step 5: During the last minute of cooking, place a slice of cheese on top of each patty, allowing it to melt. Simultaneously, toast your burger buns lightly on the grill or in a separate pan.
Step 6: Assemble the burgers: Place a cheesy patty on the bottom half of each toasted bun. Top with a generous spoonful of chopped green chiles.
Step 7: Add any optional toppings such as mayonnaise, mustard, lettuce, tomato, or onion. Crown with the top bun and serve immediately.