Melt the butter in a large, heavy saucepan or Dutch oven over medium heat. Add 2 cups of the onions, 3/4 teaspoon of the salt, and 3/4 teaspoon of the black pepper. Cook, stirring, until the onions are soft, about 4 minutes. Add the cabbage and sugar and stir until the cabbage is slightly wilted, about 3 minutes. Add the mustard and garlic and stir to mix. Add the water, stir, cover, and cook for 20 minutes, stirring occasionally. Add the cream, stir to mix, and cook, covered, for 10 minutes. Remove from the heat and keep warm.
In a large nonstick skillet over medium-high heat, heat the olive oil. Add the remaining 1 1/2 cups onions, the remaining 1 teaspoon salt, and the remaining 1 teaspoon black pepper. Cook, stirring, for 2 minutes. Add the potatoes and spread evenly over the onions. Cook, flipping the potatoes about every minute and gently mashing them, until golden brown, about 8 minutes.
Add the corned beef and cook, stirring and pressing it into the hash browns, for 2 minutes. One by one, crack the eggs over the mixture, placing them about 1/4 inch apart, reduce the heat to low, cover, and cook until the eggs are set, 7 to 8 minutes.
To serve, spread about 1/2 cup of the cabbage in the center of each serving plate and top with a wedge of corned beef hash with an egg. Sprinkle with the parsley and serve.
Yield: 8 servings