Heat a heavy skillet over medium-high until very hot. Pat the steak dry, brush with a little olive oil, and season all over with salt and pepper.
Sear the steak for 2 to 3 minutes per side until deeply browned. Reduce the heat to medium-low, add the butter, and spoon the melted butter over the top for 1 to 2 minutes, until the steak reaches your preferred doneness.
Transfer the steak to a plate to rest for 5 minutes. Meanwhile, heat a second skillet over medium with a thin film of olive oil. Crack in the eggs.
Spoon a little hot oil over the egg whites to help them set, then cover the pan for 30 to 60 seconds until the whites are opaque and the yolks are still glossy.
Slice the steak against the grain. Serve with the eggs, sprinkle with parsley, and enjoy right away.
Video
Notes
Doneness cues:
Rare: pull at 120 to 125°F; bright red center
Medium-rare: pull at 130 to 135°F; warm red-pink center
Medium: pull at 140 to 145°F; rosy center
Medium-well: pull at 150°F; mostly brown with a hint of pink