Peel potatoes; cut into quarters. Transfer to a bowl of cold water; rinse to remove starch. Drain. Transfer to a large pot of cold water; very generously salt the water. Bring to a boil over high heat. Reduce heat; simmer until potatoes are soft and the tip of a knife goes in easily, 20 to 25 minutes.

Meanwhile, cut butter in chunks and let come to room temperature.

Drain potatoes thoroughly; return to the pot over medium-low heat. Mash with a potato masher until potatoes are broken up; season with salt and black pepper. Add ¼ softened butter chunks; mash until butter just about disappears.

Add ¼ more butter; mix with a wire whisk until it just about disappears. Whisk in remaining ½ butter in 2 additions; potatoes will be light and creamy after all butter is mixed in.

Whisk in hot milk until incorporated. Season with cayenne and salt if needed. Serve immediately.