In a large bowl, mash 1 cup of the potatoes with a fork. Add the remaining potatoes, corned beef, cooking liquid, onion, garlic, mustard, parsley, thyme, and nutmeg. Season generously with pepper and mix well. Store in the refrigerator overnight or at least 3 hours.
Set up for poaching the eggs: Combine the water, vinegar, and salt in a large skillet and bring to a gentle simmer.
Preheat a large well-seasoned cast-iron skillet over medium-high heat. Add 1/4 cup of the butter and heat. When the foaming subsides, add the hash mixture and cook, stirring, for 30 seconds. Using a spatula, press the mixture down into a cake the size of the skillet. Cook, shaking the skillet occasionally, until the hash begins to brown, about 4 minutes. Reduce the heat to medium and continue cooking, shaking the skillet to loosen the hash 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. Return the skillet to the medium-high heat; add 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.
While the corned beef hash is cooking, crack an egg into a cup and carefully slide it into the hot poaching liquid. Quickly repeat with all the eggs. Poach the eggs, turning them occasionally with a spoon, until the whites are firm, or to the desired degree of doneness, about 3 to 5 minutes. Using a slotted spoon, remove the eggs and transfer to a kitchen towel. Lightly dab the eggs with the towel to remove any excess water.
Divide the hash among 4 plates and top with the poached eggs. Serve immediately.
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