Increase the heat to medium-high and add the potatoes to the pan. Cook, stirring often, for about 5 minutes. Add half of the water to the pan and use a wooden spoon to scrape any bits stuck to the bottom of the pan. Add the remaining water and cover the pan. Reduce the heat to medium-low and continue cooking, stirring often, until potatoes are tender, about 20 more minutes. Season with salt & pepper to taste.
Make 5 indentations in the hash and crack an egg into each. Cover the pan and cook on low until the eggs are set and cooked to your preference, about 8 minutes for runny yolks.