The green chile adds a touch of exciting flavor to these burgers.
Light a grill. Gently form the beef into 4 thick patties and season with salt and pepper. Grill over moderately high heat until lightly charred on the outside and medium-rare within, about 5 minutes per side. Just before the burgers are done, top each one with a slice of cheese and let it melt.
Meanwhile, heat a large cast-iron skillet and brush it lightly with oil. Crack the eggs into the skillet and cook sunny-side up over moderate heat, about 4 minutes. Spread the cut sides of the buns with the melted butter and toast on the grill. Transfer the burgers to the buns. Top with the Green-Chile Relish and the fried eggs and serve.