Bring a saucepan of salted water to a boil. Cook the potatoes until they are tender and easily pierced by a fork. Drain well. Mash the potaotes with a fork or ricer, stir in the butter and scallions. Season with salt and pepper to taste. Evenly form the mixture into patties.
Heat half of the oil in a non-stick pan or skillet over medium heat. Once hot, fry the potato pancakes on both sides until crispy and deep golden brown. Set aside to drain any excess grease.
Heat a non-stick pan over medium heat. Add the remaining oil. Crack the eggs into the pan, season with salt and pepper. Cook until the whites start to become firm, then add 1 tablespoon of water and cover with a lid to help firm up the whites of the egg. Season with salt and pepper.
Serve the eggs with potato pancakes, toast (buttered if desired, and apple.