Bring a pot of well-salted water to a boil. Add the potatoes and boil until about 3/4's of the way done, about 4 minutes. Drain well and spread out on a baking sheet to cool.
Heat a large ovenproof skillet over high heat. When hot, add the olive oil, and then add the potatoes in a single layer. Lower the heat to medium and cook, tossing occasionally, until the potatoes are crusty and browned, 10 to 12 minutes. Season with salt and pepper. Add the onion and cook until the onions brown slightly, about 2 minutes
Off the heat, make 8 small evenly-spaced wells in the hash and pour a small bit of olive oil in each well. Break an egg into each well. Sprinkle the eggs with salt and pepper. Scatter the oregano and the cheese over the hash.
Transfer the skillet to the oven and bake until the eggs are cooked to your taste, about 5 minutes for firm whites and soft yolks. Serve immediately.