Place 1 chicken breast onto a large cutting board between 2 pieces of plastic wrap. Use a meat mallet to pound it out to about ¼ to ½-inch thick. Repeat this process with the remaining 3 chicken breasts.
Lay the pounded chicken breasts out flat on a large cutting board and season with ½ teaspoon salt and ¼ teaspoon black pepper.
Place 2 ounces of Swiss cheese on each chicken breast, and top each with 2 ounces of ham. Starting with the smaller end of each chicken breast, roll it up into a tight log. Roll each log up tightly in plastic wrap, twisting the ends to close. Place the chicken logs onto a plate and refrigerate 30 minutes to 1 hour.
COOK THE CHICKEN:
When the chicken is fully chilled, get ready to bread and cook it. Preheat the oven to 350°F. Line a baking sheet with foil.
Put the flour into a shallow bowl. Lightly beat the eggs in a separate shallow bowl. To a third shallow bowl, stir together the panko breadcrumbs, remaining ½ teaspoon salt, remaining ¼ teaspoon black pepper, garlic powder, and onion powder.
To bread each chicken log, unwrap it from the plastic wrap. Dredge it in flour, shaking off the excess. Coat it in egg, letting the excess drip off. Lastly, roll it in the panko mixture.
Add about ½ an inch of oil to a large, deep frying pan. Heat over medium heat until the oil reaches 350°F. Add the breaded chicken to the hot oil (working in batches if necessary so you don’t overcrowd the pan), and cook until the chicken is golden on all sides. You can turn the heat down if necessary to keep the oil at around 350°F. This takes about 3 turns in the oil, and about 5 minutes. Once the chicken is golden, place all 4 pieces onto the prepared baking tray and bake at 350°F until they reach an internal temperature of 165°F, about 15 to 20 minutes.
MAKE THE SAUCE:
Add the butter to a saucepan over medium heat. Once melted, whisk in the flour and cook 1 minute, whisking constantly. Whisk in the milk. Turn down the heat to medium-low, and whisk in the Dijon, garlic powder, onion powder, salt, and black pepper. Bring up to a simmer, whisking constantly. Add half of the gruyere and whisk to combine. Whisk in the remaining half of the gruyere. Once fully incorporated, whisk in the parmesan.
SERVE:
Garnish the Chicken Cordon Blue with parsley and serve along with the Cheesy Dijon Sauce to drizzle on top.
Notes
The sauce reheats well over low heat on the stovetop, whisking constantly.