Put the potatoes in a large pot and cover by at least 2 inches of cold water. Season the water with about ¼ cup of salt and bring to a simmer over medium-high heat.
Adjust the heat as necessary to maintain a very gentle simmer, and cook for about 20 minutes, until tender enough to puree.
Drain the potatoes in a colander and let them steam to evaporate excess water for 1 to 2 minutes. Then quickly peel them.
Meanwhile, heat the cream in a heavy saucepan: keep warm.
Set a food mill fitted with the medium blade over a pot. Add about one-quarter of the potatoes, top with 2 pieces of the butter and one-quarter of the garlic, and puree. Repeat with the remaining potatoes, butter and garlic in 3 batches. (The potatoes can be pureed up to 3 hours ahead and held at room temperature.)
To serve, warm the potatoes in a saucepan over medium heat. As they heat, whip the cream into the potatoes.
Season to taste with salt and pepper. Transfer to a serving bowl, sprinkle with chives, and top with a spoonful of butter.