Place eggs in a pot and cover with cold water. Bring to a boil over medium-high heat. Once boiling, turn off the heat and let eggs sit in hot water for 10 to 12 minutes. Transfer eggs to cold water to cool, then peel.
In a saucepan, melt butter over medium heat. Stir in flour and cook until the mixture is golden, about 2 minutes.
Gradually whisk in milk, ensuring there are no lumps. Cook until the sauce thickens, about 5 minutes.
Stir in mustard, and season with salt and black pepper to taste.
Cut peeled eggs in half and arrange on a serving plate. Pour the mustard sauce over the eggs.
Garnish with chopped parsley and serve warm.