Weigh out the ground beef into 4oz portions and form into balls. Place in a pan and cover with foil until ready to grill. Sit aside. There's no need to pre-season.
Separate the hamburger bun halves and spread the inside of each half with butter. Toast the insides of each bun in a hot pan or griddle until golden brown.
Heat the green chile in a small sauce pan and keep warm until the burgers are ready. Add a couple of tablespoons of water to the pan periodically to maintain a spoonable consistency.
Heat a flat top grill or a cast iron skillet over high heat. Place two balls of the prepared ground beef on the surface. Let them sit undisturbed for 10-20 seconds. The slight browning of the meat at this stage will prevent the meat from sticking to the spatula during the initial smash.
Gently roll the balls over so the slightly browned side is facing up. Using firm pressure, smash each patty into the surface of the flat top or the pan using a spatula or weighted press. Hold for a few seconds before removing.
Let patties sear for 2-3 minutes.
Using a spatula with a thin edge, carefully work the spatula between each patty and the cook surface, and flip. Cook another 2 minutes. Once again lightly pressing for a few seconds.
Place a slice of cheese on top of one patty. Then stack the other patty on top of the cheese. Add a couple of tablespoons of the green chile, then top with a second slice of cheese. Squirt about a tablespoon of water near the base of the stacked patties and cover with a metal lid. The water will create steam and help the cheese melt quickly. Lift the lid to reveal your masterpiece.
Remove the double stacked patties to a pan and cover with foil to keep warm.
Assemble burgers as desired.