How to Make It
Preheat the oven to 400°. Spread the sunchokes and potatoes on 2 separate baking sheets. Toss the sunchokes with 2 tablespoons of the olive oil and season with salt and pepper. Toss the potatoes with 1 tablespoon of the olive oil and season with salt and pepper. Roast until fork-tender, 30 minutes for the potatoes and 35 minutes for the sunchokes.
Transfer the sunchokes and potatoes to a large bowl and mash with a wooden spoon or potato masher. Stir in the cheese and scallions and season the hash with salt and pepper.
In a large nonstick skillet, melt 1 tablespoon of the butter in 1 tablespoon of the olive oil over moderate heat. Scoop two 3/4-cup measures of the hash into the skillet and flatten into 4-inch rounds. Cook over moderate heat until golden and crisp, 3 to 4 minutes per side. Transfer to plates and keep warm. Repeat with another 1 tablespoon each of the butter and olive oil and the remaining hash.
Wipe out the skillet and heat the remaining 1 tablespoon of butter in it. Add the eggs and cook over moderate heat until the whites are set and the yolks are slightly runny, about 3 minutes. Serve the hash topped with the eggs.