Make the béchamel: heat milk and cream in a saucepan until steaming. In a separate pan, melt 1.5 tablespoons butter, add flour, and cook for 3 minutes, stirring constantly.

Gradually whisk the hot milk mixture into the roux until smooth. Season with nutmeg, salt, and white pepper. Cook until thickened to a spreadable consistency, then remove from heat.

Spread half the béchamel over the bread slices. On two slices, layer Gruyère cheese, spread Dijon mustard, add ham, then more cheese. Close the sandwiches.

Melt butter in a skillet over medium heat. Pan-fry sandwiches for 2 minutes per side until golden brown.

Preheat oven to 400°F (200°C). Transfer sandwiches to a baking tray. Spread remaining béchamel on top, then sprinkle with shredded Gruyère and Parmesan.

Bake for 15 minutes, then broil for 3 minutes until the top is golden, bubbling, and irresistible. Serve immediately.