Divide the beef into 4 portions and shape loose patties 1 inch wider than the buns with a shallow dimple in each center.
Season both sides with the salt and pepper and keep the patties cold while the skillet heats.
Toast the buttered buns cut-side down in a cast-iron skillet over medium-high heat for 1 minute, then remove them.
Cook the patties for 4 to 5 minutes on the first side until a dark crust forms, then flip without pressing.
Pile 1/4 cup drained green chile on each patty, cover with cheddar and cook for 4 to 5 minutes until the beef reaches 160°F and the cheese melts.
Spread mayonnaise over the buns and build each burger with lettuce, tomato and one green chile patty.
Serve at once, keeping the chile on top of the cheese so its juices stay away from the bottom bun.