Melt unsalted butter in a small saucepan over medium heat. Let it become bubbly, and then add the flour. Stir constantly for 2 to 3 minutes to eliminate any raw flour taste.

Pour in the milk gradually while whisking to prevent lumps. Continue cooking and whisking until the mixture thickens, about 2 to 5 minutes.

Whisk in dry mustard, ground ginger, gruyere cheese, and parmesan. Cook until the cheeses melt, about 3 to 4 minutes. Season with salt and pepper, then set aside.

Preheat your broiler. Spread mustard on one side of each slice of sourdough. Layer smoked ham on top.

Arrange the ham-topped toasts on a baking sheet. Ladle about 60 millilitres of mornay sauce over each slice, making a well in the center. Nestle an egg yolk into each well.

Broil the toasts until the sauce is bubbling and the egg yolks remain soft, about 3 to 4 minutes. Season lightly with additional salt and pepper. Serve immediately.