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 skillet over medium-low. Melt a tablespoon of butter in pan. Break 4 eggs into 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 warmed plates.
Cut the steak on the diagonal into thick slices. Fan the steak slices on the plates and serve immediately.