In a medium saucepan over medium heat, melt 4 tablespoons of butter until bubbling. Whisk in ¼ cup of flour to create a roux, cooking for about 2 minutes until it turns a light golden color. Gradually whisk in 2 cups of milk, stirring continuously for about 5-7 minutes until thickened. Stir in 1 tablespoon of Dijon mustard and a pinch of nutmeg, then remove from heat.

Layer approximately 6 cups of slightly stale brioche slices in a greased 9x13 inch baking dish. Pour half of the béchamel sauce over the bread, ensuring even coverage. Sprinkle 1 cup of shredded Gruyère cheese and 1 cup of diced turkey ham on top, followed by another layer of brioche. Pour the remaining béchamel sauce over this, letting it soak into the bread, then top with an additional cup of Gruyère cheese.

Using a spoon, create 4 to 6 wells gently in the top layer of the casserole. Carefully crack one egg into each well, allowing the yolks to sit on top. Sprinkle a bit more Gruyère cheese over the casserole and season with salt and pepper to taste.

Preheat your oven to 350°F (175°C). Once heated, place the casserole in the oven and bake uncovered for about 30-35 minutes, or until the top is golden and bubbly, and the egg whites are set but the yolks remain runny.

Remove the Croque Madame Casserole from the oven and let it cool for about 5 minutes before serving. Enjoy this decadent dish warm.