Use a large chef’s knife to peel the winter squash. I find it easiest to cut off the top and bottom and then carefully slice down the side of the squash. Once it’s peeled, quarter the squash and scoop out the seeds. Then grate the squash.
Peel and grate potato as well and mix potato with squash. It should be about even parts potato and squash.
Heat a few tablespoons of oil in a large skillet or griddle over medium-high heat. Add hash in an even layer. Cook for 5 minutes and flip.
Cook for an additional 5 minutes on the second side. If at any point the hash looks very dry, add another drizzle of olive oil.
Cook hash browns until they are browned nicely, then turn the heat down to low and keep them warm until the eggs are done.
For eggs, melt butter in a small nonstick pan or on a griddle over medium heat. Crack eggs and cook for about 2 minutes until the whites are mostly firm. Flip the eggs and cook for just 15-20 seconds on side two.
Serve hash with two eggs on top. Sprinkle with salt and pepper and sprinkle on some paprika and fresh scallion or chives.