Measure all ingredients and prepare two large pots and one small saucepan for the sauce. Peel the potatoes and wash them thoroughly under cold water. Halve large potatoes or cut them into similarly sized pieces so they cook evenly. Place the potatoes in a pot and cover them with cold water. Season the potato water with salt and bring it to a boil. Cook the potatoes over medium heat for approximately 18 to 20 minutes, until easily pierced with a knife.

Meanwhile, carefully place the eggs in a second pot and cover them with cold water. Bring the water to a boil and cook the eggs for about eight to ten minutes, depending on the desired doneness. Drain the cooked eggs and immediately rinse them under cold water. Allow the eggs to cool for a few minutes, then peel them carefully.

Peel the onion and cut it into very fine dice. Wash the parsley, shake it dry and finely chop the leaves.

Melt the butter in a small saucepan over medium heat. Add the diced onion to the melted butter and sauté for two to three minutes until translucent. Sprinkle the flour evenly over the onions. Whisk the flour and butter together and cook for about one minute without allowing the roux to brown. Gradually pour in the milk in small amounts while whisking continuously. Gradually add the vegetable stock while continuing to whisk vigorously to prevent lumps.

Bring the sauce to a boil, then gently simmer it over low to medium heat for about five minutes. Stir regularly while simmering to prevent the sauce from sticking to the bottom of the pan. Stir the mustard thoroughly into the hot sauce. Add the sugar and lemon juice. Season the mustard sauce with salt, black pepper and, if desired, a small pinch of nutmeg. If the sauce is too thick, stir in a little additional milk or stock. If it is too thin, simmer it for another one to two minutes.

Halve the peeled eggs or add them whole to the mustard sauce. Warm the eggs in the sauce over low heat for two to three minutes without allowing the sauce to boil vigorously.

Drain the cooked potatoes and allow them to steam dry briefly. Mix some of the chopped parsley into the potatoes or directly into the mustard sauce. Divide the boiled potatoes between warmed plates. Arrange the mustard eggs alongside and spoon over plenty of the creamy mustard sauce. Sprinkle with the remaining parsley and serve the dish immediately while hot.