Now is the right time to perfect your breakfast (or brunch) game.
Fill a large bowl with cold water. Using a food processor fitted with a shredding disk, shred the potatoes and carrots; transfer to the bowl.
In a large skillet, heat 1 tablespoon olive oil over medium heat. Add the onion and cook until softened, about 5 minutes. Add the mushrooms and cook for 3 minutes; season with salt and pepper. Add the chorizo and cook until lightly browned, about 5 minutes. Transfer to a plate.
Add 3 tablespoons olive oil to the same skillet. Drain the potatoes and carrots and arrange half in an even layer in the skillet; season with salt. Spoon the chorizo mixture on top in an even layer, cover with the remaining potatoes-carrots mixture and season with salt. Cover with foil and cook, turning the hash halfway through, until the vegetables are tender and just browned, about 12 minutes. Divide among 4 dinner plates.
In a large nonstick skillet, heat the remaining 2 tablespoons olive oil over medium heat. Add the eggs and fry for about 3 minutes; season with salt and pepper. Serve over the chorizo hash.