Prepare a baking sheet with parchment paper or silicone liner and set aside.
In a 12-inch skillet add butter and melt over medium heat.
Once the butter is melted whisk in the flour and continue whisking until the mixture is smooth and free of flour clumps.
Slowly whisk in the milk and keep whisking until smooth and the mixture thickens.
Remove from the heat and whisk in 1/2 cup Swiss cheese, fresh thyme leaves, kosher salt, ground pepper, and nutmeg.
Lay out 4 slices of bread, spreading each slice with equal amounts of Dijon mustard and 1 tablespoon of Swiss cheese sauce.
On two slices of the bread, top with four slices of Canadian bacon and two tablespoons of shredded Swiss cheese, each.
Place the Canadian bacon and cheese topped slices to your prepared baking pan. Place under the broiler until the cheese is melted, browned and bubbly.
Remove from the oven and top with the remaining slice of bread, cheese sauce side down. Top each sandwich with equal parts Swiss cheese sauce and remaining shredded Swiss cheese.
Place back under the broiler and cook until the cheese is fully melted and bubbly.
When sandwich is fully cooked top with egg and enjoy.