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To make the meatballs: In a medium bowl, use a large spoon or your hands to mix the ground beef with the egg, flour, ketchup, salt, and pepper. Cover and set aside for about 15 minutes.
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To make the mash: Bring a pot of water to a boil. Peel the turnip and cut it into 2” cubes. Remove the large stem and any leaves from the cauliflower and cut the remainder into florets.
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Add the vegetables to the boiling water and cook for about 15 minutes, or until fully softened. Drain thoroughly.
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Place the vegetables, milk, and ghee in a large mixing bowl. Use an electric mixer on high speed to beat until smooth; you don’t have to worry about overbeating like you might with potatoes. Season to taste with salt and pepper, and set the mash aside to keep warm while you finish the meatballs.
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To make the gravy: Bring the stock, mustard, and vinegar to a simmer in a small saucepan. Combine the flour with enough of the hot liquid from the pan to turn it into a thin/pourable paste; whisk this paste back into the hot liquid. Simmer, stirring occasionally, until the gravy thickens slightly, about 5 minutes. If desired, strain the gravy through a fine-mesh strainer or sieve. Set it aside.
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To finish the meatballs: Form the meat mixture into 2” meatballs. Heat the oil in a large skillet over medium heat. Brown the meatballs on all sides, about 2 minutes per side; do this in batches if necessary. Remove the meatballs from the pan and set them aside.
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Add the onions and mushrooms to the hot pan. Sauté for 3 to 4 minutes, stirring occasionally, until the vegetables begin to soften. Stir in the meatballs and gravy, reduce the heat to low, and cover. Simmer over low heat until the meatballs are cooked through, about 6 to 7 minutes. Add more beef stock if the gravy thickens too much.
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Season the meatballs and gravy to taste with salt and pepper, sprinkle with chopped fresh parsley, and serve with the turnip and cauliflower mash.
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Refrigerate any leftovers in a covered container for several days; freeze for longer storage.