Clean the leek, halve lengthways and cut into half rings 1/2 cm thick. Peel the potatoes and cut them into large pieces. Put the potatoes in a saucepan with half of the leek. Cover with salted water and bring to the boil, covered. Cook on medium heat for 20 minutes.
In the meantime, warm the milk with 1 tablespoon butter. Season with a little salt, pepper and freshly grated nutmeg.
Melt 1/2 tablespoon butter in a pan. Fry the rest of the leek in it over a medium heat for 5 minutes, season with salt and pepper.
Drain the potato and leek mixture and let it steam off briefly. Finely mash with a potato masher. Stir in the milk-butter mixture and fried leek. If necessary, season the puree with salt, pepper and nutmeg.