In a small bowl mix the gremolata ingredients.
Lightly brush the steaks on both sides with oil and season evenly with 1 teaspoon salt and ¾ teaspoon pepper. Allow the steaks to stand at room temperature for 15 to 30 minutes before grilling.
Prepare the grill for direct cooking over high heat (450° to 550°F).
Brush the cooking grates clean. Grill the steaks over direct high heat, with the lid closed as much as possible, until cooked to your desired doneness, 6 to 8 minutes for medium rare, turning once or twice (if flare-ups occur, move the steaks temporarily over indirect heat). Remove from the grill and let rest for 3 to 5 minutes.
While the steaks rest, prepare the eggs. In a large nonstick skillet over medium heat, melt the butter. Crack the eggs into the skillet, being careful not to break the yolks. Season with salt and pepper. Cook until the whites are set, 3 to 5 minutes.
Cut the steaks into thin slices. Arrange a slice of toasted bread (if using) on each serving plate, place some sliced steak on the toast, lay the eggs over the meat, and sprinkle the gremolata on top. Add a bit of hot pepper sauce, if desired. Serve warm.
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