Simmer the vegetables in an open pot in 200 ml water for about 8 minutes. Stirring occasionally. Hard boil the eggs.
Melt the fat in a saucepan, sweat the flour in it and add the stock and milk, stirring constantly. Bring everything to the boil again. Let simmer for 5 minutes. Season to taste with salt, mustard, nutmeg and pepper.
Keep the peeled eggs warm in the mustard sauce. Sprinkle with chopped parsley.