Melt butter in a medium saucepan over medium heat. Whisk in flour and cook, stirring constantly, for about 3 minutes. Gradually add milk, stirring well until the mixture is smooth. Cook, stirring, until sauce is thickened. Season with a little bit of salt, pepper, and nutmeg. Set aside while you make the sandwiches, or make the sauce up to 1 week in advance.
Assemble sandwiches:
Generously spread the bread slices with the soft butter on one side (outside). And spread the other side of béchamel (inside side).
Cut the ham to the size of the bread slices, and finely chop the mushrooms.
Arrange on four slices of bread (bechamel side) a slice of Emmental, mushroom slices, ham, and finish with a slice of Emmental. Cover everything with the other slices of white bread (béchamel below).
Bake them for 10 to 12 minutes in an oven preheated to 200°C / 392°F.