-
1
Melt butter in a medium saucepan over medium heat. Whisk in flour and cook, whisking constantly, 2 to 3 minutes to cook off raw flour taste. Pour in milk in a slow, steady stream, whisking constantly. Continue to cook, stirring until mixture thickens enough to coat the back of a wooden spoon, 4 to 5 minutes.
-
2
Whisk in dry mustard, ginger, Parmesan, and 2/3 cup of Gruyere and continue to cook, whisking, until cheeses melt and the sauce is smooth, 1 to 2 minutes. Season with salt and pepper.
-
3
Preheat broiler. Spread whole-grain mustard onto one side one side of four toast slices, dividing evenly. Top with ham and remaining 1/2 cup Gruyere, dividing evenly. Top with remaining four toast slices.
-
4
Transfer sandwiches to a baking sheet and top with Mornay sauce, dividing evenly. Broil until sauce is hot and bubbling, 4 to 5 minutes.
-
5
While sandwiches are cooking, heat butter in a nonstick skillet over medium-high heat. Crack two eggs into separate small bowls then transfer, one at a time, to skillet and cook for 2 to 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. Using a spatula, turn eggs and cook for 30 seconds to one minute more. Remove to a plate. Season with salt and pepper. Repeat with remaining two eggs.
-
6
Top sandwiches with eggs. Serve immediately.
