Mix the patty ingredients, including ½ teaspoon salt and ¼ teaspoon pepper, and then gently form four patties of equal size, each about 1 inch thick. With your thumb or the back of a spoon, make a shallow indentation about 1 inch wide in the center of the patties to prevent them from forming a dome as they cook. Refrigerate the patties until ready to grill.
In a food processor fitted with a metal blade, finely chop the parsley, garlic, and oregano. Transfer to a bowl and stir in the oil, red pepper flakes, 1 teaspoon salt, and 1 teaspoon black pepper.
Prepare the grill for direct cooking over medium-high heat (400° to 500°F).
Lightly season the patties on both sides with salt and pepper and brush the bread slices on both sides with oil. Grill the patties over direct medium-high heat, with the lid closed, until cooked to medium doneness (160°F), 9 to 11 minutes, turning once. Halfway into grilling time, place a cast-iron skillet on the cooking grates or on a side burner set to medium-high to preheat. Remove the patties from the grill.
Add the butter to the skillet. Fry the eggs sunny-side up, with the lid open, until the whites are crisp and lacy on the edges and the yolks are partially runny, 4 to 6 minutes. Toast the bread on the cooking grates over direct heat, about 1 minute, turning once.
Add the vinegar to the chimichurri. Mix well.
Top each bread slice with a patty, a fried egg, and chimichurri. Serve immediately.
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