Preheat the oven to 375 degrees F.
Use the extra softened butter to grease a 9x13-inch baking dish.
Evenly season the chicken cutlets with 1 teaspoon of the salt and 1/2 teaspoon of the pepper.
In a large skillet over medium heat, add 1 teaspoon of the cold butter and 4 of the seasoned chicken cutlets and cook until they are browned, about 2-3 minutes per side. The meat will not be cooked through.
Transfer the browned cutlets to a clean plate and repeat the cooking process with 1 teaspoon of the cold butter and the remaining chicken. Avoid wiping the skillet clean unless the butter or browned bits in it have turned dark brown.
In the same skillet, add the ham pieces and cook, stirring occasionally, until they are lightly browned, about 2 minutes.
Transfer the ham to a clean plate and set it aside.
In the same skillet, melt 3 tablespoons of the cold butter.
Gradually whisk the flour into the melted butter and cook, while stirring, for 1 minute.
Gradually whisk the milk, the chicken broth, and the cayenne into the flour mixture and bring to a simmer.
Cook the sauce mixture, stirring often, until it has thickened, about 3 minutes.
Transfer the skillet from the heat and add the Dijon mustard, and 1/2 cup of the Swiss cheese and 1/2 cup of the mozzarella cheese to the sauce mixture and stir until the cheeses melt.
Season the sauce mixture with the remaining the salt and the remaining pepper.
In a small bowl, combine the panko, 1 tablespoon of the parsley, and the remaining 1 tablespoon of the melted butter and mix until it is well-combined.
Add 1 1/2 cups of the cheese sauce to the bottom of the prepared baking dish.
Arrange the browned cutlets, overlapping them slightly, over the cheese sauce.
Evenly spread the remaining cheese sauce over the browned cutlets, then top with the cooked ham pieces.
Top the ham layer with the remaining Swiss cheese and with the remaining mozzarella cheese.
Evenly sprinkle the panko mixture over the cheese layer.
Bake the cordon bleu bake until the chicken cutlets reach an internal temperature of 165 degrees F and the casserole is golden-brown and bubbly, about 25-30 minutes.
Transfer the cordon bleu bake from the oven and allow it to cool, about 10 minutes.
Serve the cordon bleu bake garnished with the extra parsley.