In a medium saucepan or skillet, melt 2 tablespoons of the butter. Whisk in the flour and cook for a minute or two. Slowly whisk in 1 cup of the milk and bring to a gentle boil. Season to taste with salt, pepper, nutmeg. Cook until thickened, about 2 minutes, then remove from the heat and stir in the mustard.

Lay out the bread and spread each piece with some of the béchamel sauce. Layer the ham and cheese on 4 pieces, then top with the other 4 pieces, béchamel side down.

In a shallow dish, whisk together the eggs and the remaining 1/2 cup of milk.

In a large nonstick skillet, melt 1 tablespoon of the remaining butter. Soak 2 of the sandwiches in the egg mixture, then transfer to the skillet. Cook until browned and the cheese is melted, turning once, about 5 minutes. Remove from the skillet and repeat with the remaining 2 sandwiches.

If desired, sprinkle with powdered sugar and serve with raspberry jam.