Preheat your oven to 400°F (200°C).
In a saucepan, melt the butter over medium heat.
Add the flour and whisk continuously for 1 minute.
Gradually pour in the milk, whisking until smooth.
Season with salt and pepper, then cook until thickened.
Spread the sauce on one side of each bread slice.
Layer ham and cheese between the slices, sauce side inwards.
Heat a skillet and toast the sandwich on both sides until golden.
Transfer to the oven to melt the cheese, about 5 minutes.
Fry an egg in the same skillet to your desired doneness.
Top the sandwich with the fried egg and garnish with parsley.