Add the milk and heavy cream to a small saucepan and warm over medium heat until steaming (do not boil).
Melt 1 1/2 tbsp butter in the saucepan, then stir in the flour to form a smooth paste. Cook 30–60 seconds.
Slowly pour in the warm milk/cream while stirring constantly until smooth. Add salt, white pepper, and nutmeg. Cook, stirring, until thick enough to coat a spoon.
Spread a thin layer of béchamel on one side of each bread slice. Layer cheese slices, ham, Dijon mustard, then more cheese. Close the sandwiches.
Melt the 2 tbsp butter in a skillet over medium heat. Toast sandwiches on both sides until golden.
Transfer to a parchment-lined sheet pan. Spread béchamel on top of each sandwich, then top with shredded Gruyère and Parmesan.
Bake on the top rack until melted and bubbly and lightly golden (or broil briefly). Rest 2 minutes, then serve.