For the green sauce: Add the parsley, oil, capers, vinegar, Worcestershire sauce, paprika and hot sauce to a food processor and pulse until smooth. Taste for seasoning and add salt, pepper and more hot sauce as needed.
For the steak and eggs: Preheat a grill to medium-high. Sprinkle the steaks with salt and pepper and rub with 1 tablespoon oil. Grill 5 minutes per side for medium-rare and 7 minutes for medium. Let rest 10 minutes before serving. Slice the steak, if desired.
While the steaks are resting, place a cast-iron skillet on the grill or side burner and add the remaining oil to the skillet. Crack in the eggs and sprinkle with the paprika, salt and pepper. Cook until the edges are crisp and golden but the yolks are still runny, 2 to 3 minutes. Grill the bread on both sides until crisp and golden, and serve with the steak, green sauce and fried eggs.