Preheat the oven to 375 degrees . Arrange the bread on a baking sheet. Spread each slice with 1 tablespoon fig spread.
Heat 1 1/2 tablespoons butter in a small skillet over medium heat. Whisk in the flour for a minute, then whisk in the milk and thicken to coat a spoon. Season the sauce with salt, pepper and nutmeg; turn off the heat and stir in the mustard. Divide the sauce among the bread slices, spreading it liberally over the fig spread. Arrange 2 slices of prosciutto over each sauced slice of bread; evenly divide the cheese, slightly mounding it up on each open-face sandwich. Bake until golden on top, about 15 minutes.
Butter a small nonstick pan and heat over medium heat. Cook the eggs to your liking--we like ours over medium. Season with salt and pepper to taste. Place a single egg on top of each croque and serve immediately.