Pre-heat the oven to 375°F. 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 two 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.