Preheat the oven 350 degrees F.
Preheat a large cast-iron skillet over medium heat until hot, about 5 minutes. Raise the heat to high and add the oil. Season the steak generously with salt and pepper. Place the steak in the skillet and cook, turning once, until well-browned, about 4 minutes per side. Transfer the steak in the skillet to the oven and cook for 5 minutes more for medium-rare.
Transfer the steak to a cutting board. Cover it loosely with foil and let rest for 10 minutes before carving.
While the steak is resting, cook the eggs. Heat 2 skillets over medium-low. Melt a tablespoon of butter in each pan. Break 4 eggs into each skillet. Season the eggs lightly with salt and pepper, and cook until the whites are just set, about 3 1/2 minutes. (If you want the yolks to be cooked through, cover, and continue cooking for 1 to 2 minutes more.) Divide the eggs among 4 warmed plates.
Cut the steak on the diagonal into thick slices. Fan the steak slices on the plates and serve immediately.
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