Place the potatoes in a pan of boiling water and cook until fork-tender. Drain off the water and return the pan to the heat to dry the potatoes. Stir the potatoes gently to help any remaining water evaporate.
Add the butter and salt to the potatoes and mix with a rotary beater or potato masher just until the butter melts. Stir in the horseradish and Gouda cheese, mixing just until combined. Adjust the seasoning if needed.
