Place potatoes into a large saucepan with water to cover. Bring to a boil and cook until tender. Meanwhile, gently fry leeks in butter until translucent.
When potatoes are done, skin them while they are still hot and cut them into bite-size pieces. Place potatoes into a stock pot with chicken stock and leeks and season to taste with salt and pepper. Cook over medium heat until simmering, then remove from heat and stir in milk. Serve immediately.