In a large saucepan, cover the potatoes with 2 inches of water. Bring to a boil and simmer until tender, about 30 minutes. Drain and peel the potatoes. Pass them through a ricer into the saucepan. Stir in the butter and 1/2 cup of the cream until incorporated.
In a small saucepan, combine the remaining 1 cup of cream with the minced garlic and bring to a boil. Stir the garlic cream into the potatoes and cook over low heat until smooth and creamy. Add the cheese by the handful, stirring it in until melted before adding more. Season with salt and serve hot. If necessary, add a few drops of hot water to loosen the potatoes.