Preheat oven to 350 degrees F.
Lay bread out on a sheet tray and drizzle with olive oil, season with salt and pepper and toast lightly. Spoon a dollop of bechamel onto each toast. Arrange 2 slices of ham on each slice of toast and nestle it into a nest shape so the egg will rest in the center. Crack egg and drop into nest on each toast. Bake in oven until eggs are just cooked but the yolk is still slightly runny, about 10 minutes. Season with salt and pepper, if needed.
Make the bechamel sauce: Melt the butter in a saucepan over medium heat. Add the flour and stir constantly to incorporate - you do not want any color. Gradually pour in the milk as you stir, and whisk out any lumps. As the mixture thickens up continue to stir until it reaches a boil - this ensures that the flour is cooked completely. Using a microplane or a fine grater, grate nutmeg into bechamel and season with salt and pepper. Set aside and keep warm.