Scrub the potatoes and boil in salted water until cooked through.
While the potatoes are cooking, wash and slice the leeks into thin rounds. If the leeks are gritty, slice them longitudinally and rinse well before slicing.
Melt the butter in a heavy pot. Toss in the leeks and season with salt and freshly ground pepper. Cover and cook on low heat until soft and tender.
As soon as the potatoes are cooked, drain, peel, and mash.
Bring the milk to the boiling point in a small pan. Beat the buttered leeks and their juices into the potatoes along with enough boiled milk to make a soft texture. Season with salt and freshly ground pepper. Serve immediately with a lump of butter melting in the center.