Melt 2 tablespoons butter in a medium saucepan over medium heat.

Whisk in 2 tablespoons flour and cook, whisking constantly, for 2-3 minutes.

Slowly pour in 1 1/4 cups milk while whisking constantly.

Continue cooking, stirring until the mixture thickens enough to coat the back of a wooden spoon (4-5 minutes).

Whisk in 1 teaspoon dry mustard, 1 teaspoon ground ginger, 6 tablespoons grated Parmesan, and 2/3 cup shredded Gruyere.

Continue cooking, whisking until cheeses melt and the sauce is smooth (1-2 minutes).

Season the Mornay sauce with salt and pepper to taste.

Preheat broiler.

Spread whole-grain mustard onto one side of four toast slices.

Top each slice with ham and remaining 1/2 cup Gruyere.

Top with the remaining four toast slices.

Transfer sandwiches to a baking sheet.

Top sandwiches with Mornay sauce.

Broil until sauce is hot and bubbling (4-5 minutes).

While sandwiches are broiling, heat 4 tablespoons butter in a nonstick skillet over medium-high heat.

Crack two eggs into separate small bowls.

Carefully transfer eggs, one at a time, to skillet.

Cook for 2-3 minutes.

Using a spatula, turn eggs and cook 30 seconds for over-easy, 1 minute for over-medium, or two minutes for over-hard.

Remove eggs to a plate and season with salt and pepper.

Repeat with remaining two eggs.

Top broiled sandwiches with fried eggs.

Serve immediately.