Make a roux: add the butter to a small saucepan and heat over medium heat until melted. Add the flour and whisk to combine. Keep whisking until bubbles form in the mixture and cook for 2 more minutes after that. The roux should stay a pale yellow color.
Make a bechamel sauce: Heat milk in the microwave for 30 seconds. Then add slowly to the roux while whisking. Bring to a boil, whisking constantly. Lower the heat and cook until thickened (2-3 more minutes). Add salt and pepper to taste, nutmeg, and parmesan cheese. Whisk to combine.
Lay out 2 slices of bread. Top each with 3 slices of ham.
Add a few tablespoons of the bechamel sauce to the top of the ham. Top with some of the Gruyere. Place the other two slices of bread on the top of each sandwich.
Spread the outsides of each sandwich with butter. In a skillet over medium heat, toast the sandwiches for a few minutes on each side until the bread is golden brown.
Place the sandwiches on a baking sheet and top with a few tablespoons of bechamel and more Gruyere cheese. Broil until the cheese is melted, bubbly, and slightly browned.
While the sandwiches are broiling, fry two eggs. Top each sandwich with an egg and chopped chives for garnish. Enjoy!