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In a high-sided skillet, cover beets and turnips with water and bring to a boil. Season with salt and cook until tender, about 7 minutes. Drain and wipe out skillet.
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Heat oil in skillet over medium-high heat. Add boiled beets and turnips and cook until turnips begin to turn golden, about 4 minutes.
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Reduce heat to medium, add onion, and cook, stirring, until tender, about 4 minutes. Stir in parsley.
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Make four wells in the hash. Crack one egg into each and top with salt & pepper. Cover and cook until whites set but yolks are still runny, 3-5 minutes. Garnish with additional parsley if desired.