Using your hands, combine beef, onion, chiles, salt, and black pepper in a large bowl until evenly incorporated. Shape into 8 (2 1/2-ounce) balls.
Heat a large cast-iron skillet or griddle over high until smoking. Add two tablespoons of olive oil. Add four meatballs, and immediately flatten to 1/4-inch-thickness with a sturdy, wide spatula. Cook until the bottoms are crisp and deep brown, about 1 minute. Flip patties, top each with one cheese slice, and cook until bottoms are well charred, and the cheese is melted, 45 seconds to 1 minute. Remove from skillet, and cover to keep warm. Repeat with the remaining two tablespoons of olive oil, the remaining four meatballs, and the remaining four cheese slices.
Preheat broiler to high with oven rack 5 to 6 inches from heat. Brush cut sides of buns with butter. Arrange buns, cut side up, on a baking sheet. Broil in preheated oven until toasted, 1 to 2 minutes. Spread each bottom bun half with 1 1/2 tablespoons mayonnaise, top with two patties, and spoon 1 1/2 tablespoons Hatch Chile salsa over the top. Cover with top bun halves and serve immediately.
Hatch Chile Salsa
Bring a large pot of water to a boil over high. Add tomatillos, and return to a boil; cook until softened, about 4 minutes. Drain well, and let cool for 10 minutes. Transfer to a food processor. Add chiles, cilantro, onion, lime juice, garlic, salt, and agave, and process until mostly smooth, about 30 seconds. Refrigerate in an airtight container for up to 2 weeks.