Start by patting the steak dry on all sides. Moisture is the enemy of a good sear.
Next, season all sides of the steak with salt and pepper. I usually mop up the salt and pepper that fell to the side with the steak just to get it all.
In a large carbon-steel or cast-iron skillet add in the oil and turn the heat to high until it begins to smoke lightly.
Hold the steak with tongs and sear the fat cap for 30 seconds. Then, turn the steak on its side and let it cook for 1 minute. Using your tongs gently pinch the steak on its sides and move it around in a circular motion for 90 seconds to create a Maillard crust.
Flip the steak over and let it cook untouched for 1 minute. Turn the heat down to medium-low and add 2 tablespoons of butter.
Once melted, baste the steak for 2 to 3 minutes or until the top is golden brown. Set the steak aside to rest on a rack for 5 to 6 minutes.
In the meantime, drain off most of the cooking fat and return the pan to the burner over low heat and let it cool down for 1 to 2 minutes.
Turn the heat back up to medium and in 1 to 2 tablespoons of butter. Once melted, add in the eggs and fry for 2 to 3 minutes or until the desired internal doneness is achieved. Season the eggs with salt and set them to the side.
Return the pan back to the burner over medium heat and cook the minced garlic cloves until fragrant, which takes 30 to 45 seconds.
Then, add in the Worcestershire, vinegar, sugar, pepper flakes, hot sauce, and water and cook for 1 minute. Turn the heat off and swirl in 2 to 3 tablespoons of cold butter until the sauce is emulsified.
Finish the sauce with green onions and serve it with the steak and eggs, and optional side of home fries.