Preheat the oven to 375°. Butter a 9-by-13-inch casserole dish.
Season the chicken cutlets evenly with 1 teaspoon salt and ½ teaspoon black pepper.
Heat a large skillet over medium heat; add 1 teaspoon butter and 4 of the chicken cutlets. Cook for 2 to 3 minutes per side or until browned.
Remove from the skillet and transfer to a plate. Repeat with remaining chicken.
Add the ham to the drippings in the skillet, and cook until lightly browned, about 2 minutes. Remove and set aside.
Place 3 tablespoons of the butter in the same skillet over medium heat, whisk in the flour and cook for 1 minute.
Slowly whisk in the milk and chicken broth, then add cayenne if desired. Bring to a simmer and cook until thickened, about 3 minutes.
Remove from heat, stir in mustard and ½ cup each of Swiss and mozzarella until melted. Season with remaining salt and pepper.
Combine panko, parsley, and remaining melted butter in a bowl. Mix well to coat.
Spoon 1 ½ cups of cheese sauce in the casserole dish. Arrange chicken cutlets overlapping slightly, then spoon remaining sauce over chicken.
Add browned ham, top with remaining cheese, and then with panko mixture.
Bake until golden brown and bubbly, about 25 to 30 minutes. Sprinkle with additional parsley and serve with rice and green beans if desired.