Put the horseradish in a saucepan with water to cover and bring to a boil; simmer for about 5 minutes, then drain and rinse with cold water. Meanwhile, peel the potatoes and rinse them; slice them thinly with a knife or mandoline.
Preheat the oven to 400°F. Layer the potatoes and horseradish in a large nonstick, ovenproof skillet or roasting pan, sprinkling the layers with salt and pepper. Dot with the butter, then add enough half- and-half to come about three-quarters of the way up to the top.
Turn the heat under the gratin to high and bring to a boil. Turn the heat to medium-high and cook for about 10 minutes, or until the level of both liquid and vegetables has dropped somewhat. Put in the oven and cook, undisturbed, until the top is nicely browned, about 10 minutes. Turn the oven down to 300°F and continue cooking until the potatoes and horseradish are tender (a thin-bladed knife will pierce them with little or no resistance), about 10 minutes more. Serve immediately or keep warm in the oven or over very low heat for up to 30 minutes.