Toss the potatoes, butter, 1 tablespoon salt and 1/2 teaspoon pepper together in a 6-quart slow cooker. Add 1 cup of water, and cook, covered, on high until the potatoes are tender, about 4 hours. (They may have browned lightly in spots, but that is OK.)
Add the sour cream and 1 1/2 cups of the warm milk to the potatoes, and smash them with a potato masher until they are mostly smooth. If they need to be thinned out, stir in the remaining 1/2 cup warm milk. Add most the Cheddar and all the horseradish, and stir to combine. Season to taste with salt and pepper. Garnish with the remaining Cheddar and the chives.
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