1. For the chimichurri: Combine all the ingredients in a medium bowl and stir to fully incorporate. Set aside. 2. For the steak and eggs: Season the steak with salt and pepper. 3. In a large cast-iron skillet, melt the butter over medium-high heat. Place the steak on one side of the skillet, leaving room for the eggs. Cook until the steak has a dark brown sear, 3 to 4 minutes per side for medium-rare. If the pan gets dry, add another tablespoon of butter. Reduce the heat to medium-low and add the eggs. Season with a pinch of salt and pepper. When the whites start to set, cover the skillet with a lid and cook for 2 to 3 minutes more. 4. Serve with the Chimichurri Sauce.