Preheat the oven to 400 degrees F. Pierce the potatoes in a few places with a fork and microwave until almost tender, about 10 minutes, turning halfway through. Let cool slightly, then peel and cut into 1/2-inch dice.
Heat the butter in a large cast-iron skillet over medium-high heat. Add the kielbasa and cook, stirring, until it starts to brown, about 3 minutes. Add the onion and caraway seeds and cook until the onion is soft, about 3 minutes.
Stir in the potatoes, then flatten the hash into an even layer, pressing gently with a spatula. Cook undisturbed until crunchy and browned on the bottom, about 5 minutes. Flip the hash in sections and cook until browned on the other side, about 5 more minutes. Remove from the heat.
Using a spoon, make 4 shallow indentations in the hash and crack 2 eggs into each. Transfer to the oven and bake until the egg whites set, about 10 minutes. Season with salt and pepper and top with the parsley and scallions.
Photograph by Antonis Achilleos
Courtesy of Food Network Magazine