In a small saucepan over medium heat, melt butter. Add flour and whisk for a few minutes to cook out the raw taste of it. Add 2 tablespoons cold milk, and whisk constantly until the mixture get thick. Then add 1 tablespoons cold milk at a time, whisking between each addition. Continue adding milk until the consistency of the sauce gets creamy and not too thin so it can be spreadable.
Transfer sauce to a small bowl. Cover with plastic wrap making sure it touches the surface to prevent skin to form; let it cool off before transferring it to the refrigerator to chill for 2 hours.
Preheat oven to 375ºF and line baking sheet with foil; set aside.
In a small bowl, combine grated Gruyère, milk, nutmeg, salt and pepper; stir and set aside.
Spread thin layer of cold béchamel on each slice, keeping a little to spread on 4 slices later. Place a couple slices of smoked meat on 4 slices of bread before adding some cheese mixture; keep some to add on 4 slices later. Place ham on top of the cheese mixture before placing the top slice. Press down and spread more béchamel sauce on top of those 4 slices. Add the remaining cheese and place sandwiches on the prepared baking sheet.
Transfer to the preheated oven and bake for 10 minutes. Turn on the broiler and leave them there until cheese is golden brown, about 3 minutes (keep an eye on them). Remove from the heat, slice and serve immediately. Makes 4 sandwiches.