Prep. Cook fresh spinach and chop or you can also cook frozen spinach per package directions
Béchamel: Preheat the broiler with the rack in the top position. In small saucepan, melt 2 tablespoons butter over high heat. Whisk in flour and cook until lightly golden. Add milk, salt, pepper, and nutmeg, and whisk constantly until mixture thickens. Remove from the heat. Add half the cheese to the sauce and stir to melt. Divide the béchamel in two portions and add the cooked chopped spinach into one of the portions of béchamel.
Sandwiches: Lay the slices of bread on a baking sheet. Divide the Dijon between 2 slices and top with turkey.
Divide half of the first divided portion of bechamel between the remaining two bread slices and top with remaining cheese. Close the sandwiches.
Set up: Preheat griddle to medium or medium high heat depending on how hot your griddle runs
Cook: Melt two tablespoons butter onto the griddle on medium heat. Add sandwiches and cook both sides until bread is golden brown and the cheese is melted. Remove from the griddle and place on a baking sheet. Keep the griddle warm.Top sandwiches with half of one of the portions of bechamel, and place under a broiler until bubbly (watch carefully - this happens quickly). Once ready, remove from the broiler and let the sandwich rest while you cook the eggs so the cheese can set up a little bit. Melt one tablespoon of butter on the griddle and cooks eggs sunny side up.
Build: Cut the sandwich in half. Add some of the warm béchamel sauce to one half of the sandwich and place an egg on top of it. Carefully place the second sandwich on top of the first sandwich and add the béchamel sauce with the spinach on top of that, and the other egg. Serve immediately.