Place the potatoes in a saucepan and cover with cold water. Bring to a gentle boil over medium-high heat and add salt to the water. Cook until the potatoes are tender when pierced with the tip of a knife (timing will vary depending on the size of the potatoes). Drain the potatoes and put them through a food mill or a fine sieve (discard skins). Return the sieved potatoes to the saucepan over medium-high heat. Add the cubed butter, little by little, continuing to move the potatoes around in the pan using a rubber spatula and adding the milk a bit at a time to achieve the desired consistency. The potatoes should be exceptionally smooth, rich, and creamy. Season to taste with kosher salt and white pepper if desired.