Preheat your oven to 400°F (200°C).

In a medium saucepan, melt 2 tablespoons of butter over medium heat. Add the flour and whisk for about 1 minute until it forms a paste.

Gradually add the milk while continuing to whisk, ensuring there are no lumps. Cook until the sauce thickens, about 5 minutes.

Stir in the Dijon mustard, salt, and pepper. Remove from heat and mix in half of the grated Gruyère cheese. Set aside.

Assemble the sandwiches by placing 2 slices of bread on a clean surface. Layer each with 2 ounces of ham and a generous amount of the cheese sauce. Top with another slice of bread to form a sandwich. Repeat with the remaining ingredients.

Heat a skillet over medium heat and melt a little butter. Grill each sandwich for about 3-4 minutes on each side until golden brown.

Transfer the sandwiches to a baking sheet and top each with the remaining Gruyère cheese. Bake in the oven for about 5 minutes, or until the cheese is bubbly and golden.

While the sandwiches are baking, fry the eggs in the same skillet until the whites are set but the yolks are still runny.

Remove the sandwiches from the oven and place a fried egg on top of each. Garnish with chopped parsley before serving.