Bring a large saucepan of salted water to a boil. Add the potatoes and boil until tender, about 20 minutes. Drain, peel, and push twice through a food mill on medium hole setting.
Place the purée in the saucepan and dry out slightly over a low heat, stirring the butter through the purée.
Add enough milk to make the purée light and airy and work it hard with a wooden spoon to make it as fluffy as possible. Be careful not to over work it or you’ll break down the starch and the purée will become very sticky and gloopy. Serve hot.