Place each chicken breast between two pieces of plastic wrap. Using a meat mallet or rolling pin, pound the chicken to an even thickness, about 1/4 inch thick.
Remove the top piece of plastic wrap. Season both sides of the chicken breasts with salt, pepper, garlic powder, and onion powder.
Lay 2 slices of ham and 2 slices of Swiss cheese on each chicken breast, making sure they are slightly smaller than the chicken to avoid overhang.
Carefully roll up each chicken breast from the narrower end, tucking in the sides as you go to keep the filling inside. Secure with toothpicks if needed.
Prepare three shallow dishes: one with flour, one with beaten eggs, and one with panko breadcrumbs.
Dredge each rolled chicken breast first in flour, shaking off any excess. Then dip it in the beaten eggs, and finally coat it thoroughly with panko breadcrumbs.
Heat the vegetable oil in a large skillet over medium heat. Once hot, add the breaded chicken rolls.
Cook the chicken rolls for 8-10 minutes on each side, or until golden brown and cooked through. The internal temperature should reach 165°F (74°C). When the chicken is done cooking, transfer them to a paper towel lined plate to absorb any excess oil.
While the chicken is cooling, prepare the Dijon cream sauce. In a medium saucepan, melt the butter over medium heat.
Stir in the flour and cook for 1-2 minutes, or until the mixture is lightly golden and bubbly.
Gradually whisk in the chicken broth and heavy cream. Bring the mixture to a simmer, whisking constantly, until it thickens, about 3-5 minutes.
Stir in the Dijon mustard and season with salt and pepper to taste. Simmer for another 2 minutes, then remove from heat.
Serve the chicken cordon bleu topped with the Dijon cream sauce and garnish with chopped fresh parsley if desired. Enjoy!