Place the potatoes and garlic in a large saucepan. Cover with 1 to 2 inches cold water and season generously with salt; the water should taste like the sea. Bring the pot of water to a boil and cook until the potatoes are fork-tender, 20 to 25 minutes.
Drain the potatoes and garlic well and pass through a food mill or ricer. DO NOT use a food processor or blender; it will result in a very sticky and rubbery end product.
While passing the potatoes, bring the heavy cream to a boil in a small saucepan and remove from the heat.
While the potatoes are still hot, add one-third of the cream and the butter and stir vigorously into the potatoes. Repeat this process with the remaining cream until you achieve a velvety and smooth consistency. Taste the potatoes for seasoning and add salt if needed. Serve immediately or cover with foil and keep warm in an oven on low heat.