Working with one chicken breast at a time, lay the meat on a cutting board and use a sharp knife to slice each breast in half horizontally into two even pieces. Cover with sheets of plastic wrap and pound each piece with a meat mallet to about ¼-inch thick with a meat mallet, if needed.

Place a slice of ham and a slice of cheese on top of each flattened chicken breast. Roll each chicken breast into a tight roulade, then wrap them tightly in pieces of plastic wrap. Refrigerate for 30 minutes to secure without toothpicks. Alternatively, you can use toothpicks to hold each roulade together and skip the refrigerating time.

Preheat oven to 350 degrees F. Combine the contents of a box of Shake N' Bake (each box comes with 2 packets and you will use them both) with the Parmesan cheese and packet of Good Seasons Garlic & Herb Dressing in a shallow dish. Stir well to combine, then set aside. Also set aside a shallow dish or plate with ¼ cup of flour for dredging, and a third shallow dish with the beaten eggs for an egg wash.

Unwrap and dredge each roulade first in flour, shaking off any excess, then in the beaten egg, then in the Shake N' Bake mixture. Repeat the process with all of the remaining chicken.

Combine the cream of chicken soup, sour cream, and chicken broth in a medium bowl. Season with salt and pepper, to taste. Pour into a 9x13-inch pan and set aside.

Heat 3-4 tablespoons of vegetable oil in a large skillet. Working in batches, brown the chicken cordon bleu on all sides for 2-3 minutes just to crisp up the outside and give it some color. Transfer to a large baking dish with the chicken cordon bleu sauce.

Bake for 40 minutes until the sauce is bubbling and the chicken is cooked through.