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Ingredients

1. 2 1/4 pounds ground chuck, preferably organic
2. Salt and freshly ground pepper
3. 1/2 pound mild American blue cheese, such as Maytag Blue Cheese, cut into 4 thin slices
4. 4 large eggs
5. 4 brioche buns, split
6. 4 tablespoons unsalted butter, melted
7. Vegetable oil
8. 1 cup Green-Chile Relish, warmed up

Directions

1. 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.
2. 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.