Peel the potatoes and place them in a large pot with cold water to cover.
Boil them for 40 minutes or until they are very soft. They are ready if they fall apart when poked with a fork. Drain in a colander.
Off the heat, put 12 tablespoons of the butter into the pot in which the potatoes were boiled.
When it begins to melt from the heat of the pot, fit a food mill with a fine blade over the pot and put the potatoes through it. They really have to be milled; if they are actually mashed, they will be lumpy.
Put the pot of potatoes back on the stove over VERY LOW heat. Stir with a whisk to incorporate the butter.
Little by litle, begin pouring in the hot milk. Stir a LOT as you pour so the potatoes get really fluffy. You may not need all of the milk; add enough so the potatoes get to the consistency of thick whipped cream.
Keep the potatoes over low heat and keep stirring. The potatoes have to be really stretched. If they get too thick, add more milk.
Just before serving, beat in the remaining 4 tablespoons of butter. Salt and pepper to taste.