Line up 8 slices of bread on a work surface.
Heat a small saucepot over medium heat. Add 2 tablespoons butter and melt. Add flour to the butter and whisk together. Cook flour and butter together for 1 to 2 minutes, then whisk in half-and-half or milk. Bring sauce to a bubble and thicken another 1 or 2 minutes. Season sauce with salt, then whisk in prepared Dijon mustard.
Heat a large nonstick griddle or skillet over medium heat.
Spread each slice of bread with Dijon bechamel sauce and build 4 sandwiches with 2 or 3 slices of ham and 2 slices of cheese per sandwich. The Dijon bechamel sauce will help hold the meat and cheese in place. Make slices of ham and cheese rest flat; do not pile the meat or cheese. Butter the tops of the 4 sandwiches and set onto griddle or grill, buttered side down. Butter opposite side of each sandwich while the first side is grilling. Toast sandwiches 4 minutes on each side, turn with thin spatula. Serve with cornichons, pickled onions and extra Dijon mustard to pass at table.