In a medium saucepan over medium heat, melt 4 tablespoons of butter. Once melted, whisk in ¼ cup of flour to create a roux, stirring constantly for about 2 minutes until it becomes slightly golden. Gradually pour in 2 cups of whole milk, whisking continuously until the sauce thickens (about 5-7 minutes). Stir in 1 tablespoon of Dijon mustard, a pinch of nutmeg, and season to taste with salt and pepper.

Preheat your oven to 375°F (190°C). In a generously greased 9x13-inch baking dish, layer 6 cups of slightly stale brioche cubes evenly across the bottom. Then, add 8 ounces of sliced turkey ham and sprinkle 1 ½ cups of shredded Gruyère cheese on top. Pour the prepared béchamel sauce over the layers, ensuring everything is evenly coated.

If you prefer to serve separate portions, preheat your oven and use individual ramekins instead. Spoon equal amounts of the brioche, turkey ham, and Gruyère mixture into each ramekin, pouring béchamel sauce over until just covered.

Place your baking dish (or ramekins) in the preheated oven and bake for about 30-35 minutes, or until the top is golden brown and bubbly.

In the last 10 minutes of baking, carefully remove the casserole from the oven. Gently crack 6 eggs on top, spacing them evenly. Return to the oven and bake for an additional 10-12 minutes, or until the egg whites are set while the yolks remain slightly runny.