Preheat oven to 375°. Use a rolling pin to flatten each bread slice. Melt 2 tablespoons butter and brush bread slices with melted butter.
Gently press each bread slice into 6 jumbomuffin cups. Set aside.
In a small saucepan, make béchamel sauce by melting remaining 2 tablespoons of butter over medium heat. Whisk in flour for 30 seconds.Slowly whisk in milk, salt, nutmeg and mustard.
Continue to whisk while mixture thickens, about 3 minutes. Remove and set aside.
Fill each muffin cup with slice of ham, sprinkle of Gruyere cheese, and tablespoon full of bechamel sauce. Top each muffin cup with an egg (yolk and whites). Sprinkle with salt and pepper, as desired.
Bake for 18-22 minutes or until whites are set and yolk is slightly set. Sprinkle with diced herbs (like fresh thyme or parsley) as desired.
Notes
You can easily turn these into Croque Monsieur breakfast sandwiches by omitting the egg on top.