Preheat the oven to 375 degrees F.
Using the extra butter, grease a 9x13-inch casserole dish.
Season the chicken evenly with 1 teaspoon of the salt and 1/2 teaspoon of the black pepper.
In a large skillet over medium heat, add 1 teaspoon of the remaining melted butter and allow it to heat.
Add 4 of the chicken cutlets and cook, flipping them once, until browned, about 2-3 minutes per side.
Transfer the browned chicken cutlets to a clean plate and repeat with 1 teaspoon of the remaining butter and the remaining chicken cutlets.
Add the ham to the same skillet and cook, while stirring occasionally, until lightly browned, about 2 minutes.
Transfer the cooked ham to the same plate as the chicken cutlets.
In the same skillet over medium heat, add 3 tablespoons of the remaining melted butter and allow it heat.
Slowly add the flour to the melted butter in the skillet, while whisking to combine, until it is incorporated and the raw flour smell disappears, about 1 minute.
Slowly add the milk and the chicken broth to the flour mixture, while whisking to combine, until incorporated and smooth.
Add the cayenne pepper to the sauce mixture and whisk to combine.
Bring the sauce mixture to a simmer and cook, while stirring often, until thickened, about 3 minutes.
Transfer the sauce mixture from the heat.
Add the mustard, 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 cheese sauce with the remaining salt and the remaining black pepper.
In a small bowl, add the panko breadcrumbs, 1 tablespoon of the parsley, and the remaining 1 tablespoon of the melted butter and mix well to combine and coat.
Spoon 1 1/2 cups of the cheese sauce into the prepared casserole dish.
Arrange the chicken cutlets and the ham pieces on top of the sauce layer in the casserole dish, overlapping the cutlets slightly.
Add the remaining cheese sauce over top of the chicken cutlets and the ham pieces, covering them evenly.
Top the cheese sauce layer with the remaining Swiss cheese and the remaining mozzarella cheese.
Add the panko mixture evenly over the top of the casserole.
Bake until the casserole is golden-brown and bubbly, and the chicken is cooked through reaching an internal temperature of 165 degrees F, about 25-30 minutes.
Sprinkle the top of the casserole with the extra parsley.
Serve with the rice and the green beans.