Whisk in flour and cook for 1 minute, until smooth.
Add milk in all at once, whisking until mixture becomes thick. Stir in salt, pepper and nutmeg. Set aside.
Assemble the sandwich: Spread 2 tablespoons butter on slices of bread. Flip bread over and top with 2 tablespoons of the sauce. Sprinkle cheese over the sauce, followed by 3 slices of ham, then sprinkle more cheese. (It will be half a cup of cheese in total per sandwich.) Top with remaining slices of bread, butter side up.
In a cast-iron pan, place sandwiches butter side down.
Grill for 2 minutes or until golden brown, flip and cook for another 2 minutes.
Turn off heat and top each sandwich with more sauce and remaining cheese. Place in the broiler to melt cheese.
Meanwhile, heat a nonstick skillet on medium heat.
Crack egg and cook sunny side up until white is set.
Once cheese is melted, remove sandwich from broiler.