Heat 3 tablespoons of the oil in a large skillet over medium-high heat. Add the cauliflower and onion and cook until browned, 10 to 12 minutes. Add the ham, season with the paprika, ¾ teaspoon salt, and ¼ teaspoon pepper, and cook, tossing, until tender, 10 to 15 minutes; transfer to a plate.
Reduce heat to medium and heat the remaining 1 tablespoon of oil in the skillet. Fry the eggs to the desired doneness, 2 to 3 minutes for slightly runny yolks. Serve on the hash and sprinkle with the parsley.
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