Preheat Oven:

• Preheat your oven to 375°F (190°C).

Prepare Chicken Breasts:

• Put each chicken breast inside two pieces of plastic wrap.
• Use a meat mallet to pound the chicken breasts to an even 1/4 inch thickness.

Stuff and Roll Chicken:

• Lay 2 slices of Black Forest ham and 2 slices of Swiss cheese on each prepared chicken breast.
• Carefully roll up each chicken breast, tucking in the sides as you go, and secure with toothpicks.

Coat Chicken:

• Dredge each stuffed and rolled chicken breast in the seasoned flour, shaking off any excess.
• Dip the floured chicken into the beaten eggs, then coat with the seasoned panko breadcrumbs.

Fry Chicken:

• In a large skillet, heat the butter and vegetable oil over medium-high heat.
• Fry the chicken breasts until golden brown on both sides, about 3-4 minutes per side.

Finish in Oven:

• Transfer the chicken breasts to a baking dish and bake in the preheated oven for 20-25 minutes, or until the chicken is cooked through and the internal temperature reaches 165°F (74°C).

Prepare the Dijon Cream Sauce:

Make Roux:

• In a small saucepan, melt the butter over medium heat.
• Add the flour and whisk continuously for about 1-2 minutes to form a roux.

Add Liquids:

• Gradually whisk in the half-and-half, chicken broth, and white wine (if using), ensuring there are not any lumps.
• Continue to whisk until the mixture thickens and comes to a simmer, about 3-5 minutes.

Finish Sauce:

• Stir in the Dijon mustard and grated Parmesan cheese.
• Continue to cook, whisking frequently, until the cheese has melted and the sauce is smooth and creamy.
• Season with salt and pepper to taste.

Serve:

• Remove the toothpicks from the chicken breasts.
• Serve the chicken cordon bleu with the Dijon cream sauce drizzled over the top.