In a large saucepan, combine the potatoes, 1 teaspoon salt, and enough cold water to cover. Bring to a boil over high heat. Lower the heat to maintain a simmer and cook until fork tender, about 10 minutes. Drain the potatoes and return to the pan. Shake the pan over medium heat for about a minute to dry the potatoes. Transfer the potatoes to a food mill, ricer, or bowl.
Put the milk and butter in the saucepan, and heat over medium-high heat until the butter melts and the milk is hot. Remove pan from the heat and add the horseradish. Mash the potatoes through the food mill, ricer, or by hand into the pan. Stir to combine with the milk and butter. Season with salt and pepper to taste. Serve immediately.
The potency of prepared horseradish can vary greatly, so adjust the quantity to taste, if needed.
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