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Melt the butter in a small saucepan over medium heat. Add the flour and stir over the heat for 2 to 3 minutes to cook off the raw flour taste. Add the milk in a steady stream while whisking, then continue to cook and whisk until the mixture thickens enough to coat the back of a spoon. Whisk in the mustard, ginger, gruyere, and parmesan. Continue to cook and stir until the cheeses melt and the sauce is smooth, 3 to 4 minutes. Season to taste with salt and pepper. Set aside.
Preheat broiler. Spread whole grain mustard onto one side of the toast slices. Top slices of the toast with the ham.
Transfer the ham-topped toasts to a baking sheet and top each with a ½ cup of the mornay sauce.
Carefully create a well atop each toast, with the ham and mornay sauce and drop each yolk into each well.
Broil the toasts until the sauce is hot and bubbling, but the yolk is still soft, 3 to 4 minutes. Lightly season with salt and pepper and serve immediately.