In a small pot, melt butter over medium heat. Add flour and stir 1 minute. Gradually whisk in milk followed by salt, pepper, and nutmeg. Bring to a simmer and continue stirring until sauce thickens in 2-3 minutes. Remove from heat and stir in cheese. Set aside.
Place 1 tablespoon of butter in a frying pan on medium heat. Place ham steaks in pan once butter has melted. Cook 3-4 minutes on each side until ham is cooked through and browned. Remove ham from pan and keep warm.
Lightly butter one side of each slice of bread and spread mustard on the other side. Place butter side down in toaster oven until toasted.
While bread is toasting, put another tablespoon of butter in the frying pan to melt over medium heat. Crack eggs and fry in pan until whites are set in 3-4 minutes. Season eggs with salt and pepper.
Assemble by laying a ham steak on each slice of bread (mustard side up) and then place the fried egg on top. Generously pour cheese sauce over the top, and it’s ready to serve.