Chop the parsley leaves. Peel the potatoes and cook as boiled potatoes. Boil the eggs hard in boiling salted water for approx. 8 minutes.
Melt the butter in a saucepan and add enough flour to make a creamy mixture. Let it foam up to a light yellow color while stirring constantly. Dilute the milk with the same amount of water and gradually add the milk-water mixture and stir in with a whisk. Bring it to the boil again and again until the sauce is as smooth as you want it to be. Add the chopped parsley and season with salt and pepper.
Peel the eggs, place them in the sauce and let them steep for 1 - 2 minutes. Drain the potatoes, steam them and serve with the sauce and eggs.