Heat the oven and flatten the chicken: Heat the oven to 400°F and line a rimmed baking sheet with parchment. Lay each chicken breast between two sheets of plastic wrap and pound it to an even 1/4-inch thickness. Season both sides with 1/2 teaspoon of the salt and the black pepper.

Layer and roll: Lay 2 slices of ham and 2 slices of Swiss on each flattened breast, leaving a 1/2-inch border all the way around. Roll each one up tightly from the short end, tucking the sides in as you go, and secure the seam with two toothpicks.

Set up the breading: Line up three shallow dishes: the 1/2 cup flour in the first, the beaten eggs in the second, and the panko stirred with 1/3 cup of the parmesan, the garlic powder, and the remaining 1/2 teaspoon salt in the third.

Bread the rolls: Roll each chicken bundle in the flour, then the egg, then press it firmly into the panko mixture on all sides. Set them seam side down on the baking sheet and drizzle the 3 tablespoons melted butter evenly over the tops.

Bake: Bake for 30-35 minutes, until the crust is deep golden and a thermometer pushed into the thickest part of the chicken reads 165°F.

Make the Dijon cream sauce: While the chicken cordon bleu bakes, melt the 2 tablespoons butter in a saucepan over medium heat. Whisk in the 2 tablespoons flour and cook for 1 minute. Slowly whisk in the chicken broth, then the heavy cream, and simmer for 3-4 minutes until it thickens enough to coat the back of a spoon.

Finish the sauce: Take the pan off the heat and whisk in the Dijon mustard and the remaining 1/3 cup parmesan until smooth.

Rest and serve: Let the chicken rest for 5 minutes, then pull out the toothpicks and slice each roll crosswise. Spoon the sauce over the top and scatter with parsley.