Prepare Béchamel: In a saucepan over medium heat, melt 4 tablespoons of butter. Once melted, whisk in ¼ cup of flour, cooking for about 1-2 minutes until golden. Gradually add 2 cups of milk while continuously stirring to avoid lumps. Cook until thickened, about 5 minutes. Stir in 1 tablespoon of Dijon mustard, a pinch of nutmeg, salt, and pepper to taste, ensuring a rich, creamy béchamel.

Layer Ingredients: Grease a 9x13-inch casserole dish with butter. Begin layering by placing 6 cups of cubed brioche at the bottom. Add 1 cup of smoky turkey ham on top, followed by 1½ cups of grated Gruyère cheese. Pour the warm béchamel sauce evenly over these layers.

Add Eggs: Create small wells in the casserole for each egg and crack an egg into each well.

Bake the Casserole: Cover the dish with aluminum foil and bake in a preheated oven at 350°F (175°C) for 25 minutes. Remove the foil and bake for an additional 10-15 minutes.

Serve and Enjoy: Let it cool for 5-10 minutes before serving. Dish out portions and pair with a light salad or seasonal fruit.