HASH: In large bowl, place 1 cup of the potatoes and mash with a fork. Add the remaining 1 cup potatoes, corned beef, cooking liquid, onion, garlic, mustard, thyme, and nutmeg. Season generously with pepper and mix well, store in the refrigerator overnight.
Remove the hash mixture from the refrigerator and stir in the parsley. Heat 1/4 cup of butter in a large non-stick skillet over medium-high heat. When the foaming subsides, add the hash mixture and cook, stirring, for 30 seconds. Using a spatula press the mixture down into a round cake the size of the skillet. Cook shaking the skillet occasionally, for 4 minutes. Reduce the heat to medium and continue cooking, moving the skillet occasionally, until the underside is browned and crusty, about 6 minutes more.
To flip the hash, set a plate the size of the skillet on top of the pan. Invert the pan so the hash falls on to the plate as an intact cake. Invert the hash onto another plate, cooked side up.
Return the skillet to the heat, raise heat to medium-high, and heat the remaining 1 tablespoon butter. When the foaming subsides, slide the hash into the skillet cooked-side up. Cook shaking the skillet occasionally, for 3 minutes. Reduce the heat to medium and cook, shaking the skillet occasionally, until the underside is browned and crispy, about 5 minutes more. Set aside covered with foil to keep warm.
EGGS: While the corned beef hash is cooking, combine the water, vinegar and salt in a large skillet and bring to a simmer. Crack the eggs into separate cups. Carefully slide the eggs into the skillet and cook, turning occasionally with a spoon, until firm, about 3 to 5 minutes. Using a slotted spoon, remove the eggs and transfer to kitchen towel. Lightly dab the eggs with a kitchen towel to remove any excess water. Divide the hash among plates and top with the poached eggs.