Cut the thick country French bread into 1-inch cubes and place them into a large mixing bowl. Preheat the oven to 350°F (175°C).

In a separate bowl, whisk together 6 large eggs, a pinch of black pepper, and fresh thyme until light and frothy for about 2 minutes. Pour the egg mixture over the bread cubes and let them soak for 10-15 minutes.

Grease a 9x13-inch casserole dish, layer the soaked bread mixture, followed by a sprinkle of grated Gruyere cheese and slices of smoked ham. Repeat layers until all ingredients are used, finishing with cheese on top.

In a saucepan, melt butter over medium heat, whisk in flour for 2-3 minutes until golden. Slowly add milk while whisking to avoid lumps until it thickens for about 5 minutes. Pour the sauce over the casserole layers.

Bake in the preheated oven for 35-40 minutes until the top is golden brown and bubbly. Cover with foil halfway through if excessive browning occurs.

If desired, fry eggs to serve on top of each portion for added richness and flavor.