Add 2 teaspoons of the butter to a very large pan over medium-low heat until melted.
Add 2 slices of bread to the pan and cook for 1 to 2 minutes or until toasted and lightly browned.
Put 2 more teaspoons of butter into the pan and flip the bread over to cook on the other side.
On two of the pieces of toasted bread add 2 teaspoons of mustard to each spreading it around to cover.
Add 4-6 ounces of ham to the top of the toast with mustard and then evenly spread/layer on as follows: 4 ounces of bechamel, 1/2 cup shredded cheese, a slice of toasted bread, and finally a ¼ cup of shredded cheese.
Place under the broiler on low heat on the middle rack for 3 to 4 minutes or until the cheese is melted and browned.
In the meantime, fry the eggs to the desired internal temperature.
Top the sandwich with the fried egg and slice the sandwich and garnish optionally with chopped parsley.
Repeat the process to make the other 5 sandwiches.
Notes
Chef notes:
Make-Ahead: This sandwich is meant to be eaten as soon as it is done.
How to Store: Cover and place in the refrigerator for up to 2 days. This sandwich will not freeze well.
How to Reheat: Place the sandwich on a cookie sheet tray lined with parchment paper and bake at 350° for 4-6 minutes or until hot. You can also reheat in the microwave until warm.
Other cheeses you can use are swiss, Emmental, Jarlsberg, Beaufort, or raclette.
You do not need to use mustard, but it does add some great flavors.
Some other great breads to use are sourdough, white, or a boule.
Another variation you can do is swap out the ham with cooked and sliced chicken breasts.
I like to serve up the Croque madame with extra mustard and sliced pickles.