Preheat your oven to 400°F (200°C).

In a small saucepan over medium heat, melt the butter. Once melted, whisk in the gluten-free flour and cook for about 1 minute until bubbly.

Gradually add the milk, whisking constantly until the mixture thickens, about 3-5 minutes. Stir in the Dijon mustard, salt, and pepper. Remove from heat.

Assemble the sandwich by placing one slice of gluten-free bread on a baking sheet. Layer the ham and half of the Gruyère cheese on top. Place the second slice of bread on top.

Pour the prepared sauce over the sandwich, ensuring it is well-coated. Sprinkle the remaining Gruyère cheese on top.

Bake in the preheated oven for 10-12 minutes, or until the cheese is bubbly and golden.

While the sandwich is baking, fry the egg in a non-stick skillet to your desired doneness.

Once the sandwich is done, remove it from the oven and top with the fried egg. Garnish with chopped parsley before serving.