For the chimichurri: Add the vinegar, olive oil, oregano, red chile flakes, garlic and parsley to a food processor. Season with salt and pepper and pulse gently until just chopped. Add half of the chimichurri to a plastic bag and reserve the rest for serving.
For the steaks: Add the steaks to the bag with the chimichurri and make sure they are coated. Seal and rest at room temperature for 2 hours.
Bring a grill pan to medium-high heat.
Remove the steaks from the bag, making sure to remove as much chimichurri from them as possible. Season the steaks with salt and pepper. Grill on one side for 2 to 3 minutes, then flip and grill until the internal temperature is 125 degrees F for medium rare, 2 to 3 minutes more. Remove to a plate, cover and let rest for 10 minutes.
For the eggs: Whisk the eggs with 2 teaspoons water in a large bowl until combined and frothy.
Add the butter to a large nonstick pan over low heat and as it melts, tilt the pan to make sure the butter coats the bottom and sides. Pour the eggs into the pan and allow them to sit until a thin layer begins to set on the bottom and sides of the pan, about 30 seconds. Using a whisk, gently scrape the bottom and sides of the pan using a twisting motion to free any eggs attached to the corners and sides of the pan. Allow the eggs to rest again for another 30 seconds, then "scramble" again with the whisk. Repeat this step until the eggs are almost done but still wet, about 6 minutes total. Season with a pinch of salt and a grind of pepper. Remove the pan from the heat and continue to "scramble" them with the whisk to finish, about 30 more seconds. Serve with the rested steaks and reserved chimichurri.
If you have trouble releasing the egg from the whisk, simply flatten the wand of the whisk and tap it on the side of the pan.