Peel the potatoes and rinse them in cold water. Cut the potatoes in fourths, place in a large pot and cover with water. Bring to a boil and cook until fork-tender, 20 to 25 minutes.
Drain the potatoes, then return them to the pot and stir to evaporate any excess moisture. Rice the potatoes into a large saucepan. Stir in the heavy cream and butter until the potatoes are smooth. Turn the heat to low and stir in the cheese, handful by handful, until fully melted. Season with salt.
Transfer to a fondue pot (or a slow cooker for larger servings), stirring occasionally to prevent scorching.