For the horseradish cream, whisk the horseradish into the sour cream and season with salt and pepper, to taste.
Heat 2 tablespoons of the butter in a large (10-inch) nonstick skillet over moderate heat. Add the onion and rosemary and cook, stirring occasionally until soft, about 10 minutes. Add the garlic to the onion and cook another minute.
Meanwhile, with the shredding blade of a food processor, or with a box grater, grate the potatoes into a colander. Rinse and pat very dry. (Giving them a turn in a salad spinner helps, too.)
Stir the potatoes into the vegetables in the skillet. Cook the potatoes, undisturbed until they get crusty on the bottom. Then, stir the crispy bottom into the tender potato, and continue to cook to re-crisp the bottom. Continue this browning and periodic stirring until the potatoes are almost cooked through, about 8 minutes. Meanwhile, shred the carrot and beet.
Remove the rosemary sprigs. Stir the carrot and beets into the potato hash, and cook until slightly soft, about 5 minutes. Press the hash into an even cake. Add the remaining 2 tablespoons of butter in bits along the sides of the skillet, and then brown the hash over high heat. Turn the hash in large sections with a spatula to brown the other sides, another minute.
Divide the vegetable hash among 4 plates and serve warm with the horseradish cream.
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