Begin by preheating your oven to 350°F (175°C). Cut the bread into 1-inch cubes and toast them in the oven for about 10 minutes.

In a large mixing bowl, combine the toasted bread cubes, Gruyere cheese, and diced smoked ham. Crack in the eggs and mix until well coated.

In a saucepan over medium heat, melt the butter, whisk in the flour, then gradually pour in the milk and cook until thickened. Add a pinch of nutmeg.

Pour the bechamel sauce over the bread mixture and stir until combined. Grease a casserole dish and transfer the mixture into it.

Create wells in the mixture and crack eggs on top. Sprinkle with thyme and black pepper.

Bake for approximately 38-40 minutes or until bubbly and golden brown. The eggs should be set but still slightly jiggly at the center.

Let the casserole cool for about 10 minutes before slicing and serving warm.