Preheat oven to 375℉.
Grease oven safe casserole dish and set aside.
Whisk together eggs and 1/2 cup milk. Season with salt and pepper. Set aside.
Trim crust from slices of bread. Use 6 of the slices to line the bottom of the casserole dish. Cut slices in half if needed to make them fit the bottom of the pan in an even layer.
Top bread with 6 slices of the deli ham. Set aside.
Melt butter in a skillet over medium heat.
Once butter has melted sprinkle in the flour. Use a whisk to stir together the flour and butter. Cook, stirring constantly for about a minute to cook off the raw flour taste.
Add 1½ cups of half & half or milk to the pan.
Add dijon mustard, freshly grated nutmeg, ½ teaspoon salt and ½ teaspoon pepper. Whisk constantly until no lumps remain and bring mixture to a simmer and it thickens, about 3-4 minutes. Remove from heat.
Spoon half of the béchamel sauce evenly over the ham in the casserole dish.
Top with the remaining 6 slices bread, cutting them to fit if needed.
Pour the reserved egg mixture evenly over the casserole dish
Top with remaining slices of ham and a layer of the remaining béchamel sauce.
Top with the grated Gruyère cheese.
Cover casserole dish with foil and bake for 30 minutes.
Remove foil and continue to cook until heated through, hot and bubbly, the cheese is completely melted, and the edges are a crisp golden brown.
Allow casserole to rest for 10 minutes before slicing.
Top with freshly chopped parsley to serve.