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.