Preheat a large skillet over medium heat. We recommend using a cast iron skillet for best results. Add 2 tablespoons of olive oil to the hot skillet, followed by the potatoes, with plenty of salt and pepper. Cook the potatoes and don't stir for about 7-10 minutes to create a nice golden crust. Toss, and cook for another 5-7 minutes, letting them sit without touching once more. Finally, toss once more, and cook (without touching) for an additional 4-5 minutes. At this point, the potatoes should be nice and golden brown on multiple sides.
Melt 2 tablespoons of butter in the skillet. Add the onion, followed by another pinch of salt. Cook until softened, about 4-5 minutes, then add in minced garlic, thyme, oregano, red pepper flakes, and smoked paprika. Cook until fragrant, about 1 minute, stirring often to prevent the garlic from burning.
Increase the heat to medium-high, then add in the corned beef, and Worcestershire sauce. Toss to coat. Cook for 5-7 minutes, stirring infrequently to render out the fat and crisp up the corned beef.
Create 4 small wells for the eggs, filling each of the wells with the remaining olive oil. Crack the eggs into the wells, then season with salt and pepper. Reduce the heat to medium-low. Cover for 4-8 minutes until the eggs are cooked to your liking. We prefer a runny yolk, so we cook the eggs for between 4 - 5 minutes. Just make sure the egg whites are set!
Garnish with more parsley, salt, and pepper. Store leftovers in an airtight container in the fridge for 4-5 days. Reheat on the stovetop, in the oven, or in the air fryer. We don't recommend reheating the eggs, and instead, making them fresh everytime.