Prepare the Béchamel Sauce: In a medium saucepan, melt 4 tablespoons of butter over medium heat until foamy. Whisk in 4 tablespoons of flour, stirring continually for about 1-2 minutes until lightly golden. Gradually pour in 2 cups of milk, whisking constantly until the mixture thickens and bubbles, about 5-7 minutes. Stir in 1 tablespoon of Dijon mustard and a pinch of nutmeg, then remove from heat.

Assemble the Casserole: Preheat your oven to 375°F (190°C). Grease a 9x13-inch baking dish and layer the bottom with 8 slices of slightly stale brioche. Sprinkle a generous layer of chopped turkey ham and 1 cup of grated Gruyère cheese on top. Repeat the layering process with the remaining brioche, ham, and cheese.

Add the Béchamel Sauce: Pour the prepared béchamel sauce evenly over the assembled layers in the baking dish. Ensure that every slice of brioche is well-coated and use a spatula to gently press down the layers.

Prepare the Eggs for Topping: In a mixing bowl, whisk together 4 large eggs and 1 cup of heavy cream until smooth. Season with salt, pepper, and a dash of nutmeg.

Bake the Casserole: Pour the egg mixture over the béchamel-coated casserole. Bake for 30-35 minutes, until the top is golden brown and bubbly.

Cool and Serve: Once baked, remove from the oven and let cool for about 10 minutes. Slice into squares and enjoy warm.