Begin by cutting thick slices of crusty, country French bread into cubes, about 1-inch in size. Set the cubes aside to air out.

In a medium saucepan over medium heat, melt a generous amount of butter. Whisk in flour until it forms a smooth roux and cook for about 2 minutes. Gradually add milk, whisking constantly until it thickens.

Beat the eggs in a large mixing bowl. Add the grated Gruyere cheese, black pepper, and fresh thyme, mixing until well combined. Fold in the bread cubes, ensuring they're well-coated. Let sit for 10 minutes.

Preheat your oven to 350°F (175°C). Grease a casserole dish and layer half of the egg-bread mixture, followed by ham and cheese. Repeat with remaining mixture.

Bake uncovered for about 35-40 minutes until golden brown and a knife inserted comes out clean.

Optional: Broil a few eggs on top for added flair just before serving. Broil for about 3-5 minutes until eggs are just set.