Preheat oven to 400 degrees Fahrenheit (200 degrees Celsius) and line a baking sheet with parchment paper.
Place chicken breasts between plastic wrap and pound to an even 1/4 inch thickness using a meat mallet.
Season chicken with salt and pepper, then layer one slice of ham and one slice of cheese on each breast.
Roll the chicken tightly starting from the small end and secure with toothpicks.
Prepare three dredging stations: flour, beaten eggs, and Panko mixed with garlic and onion powder.
Dredge each roll-up in flour, dip in egg, then press into Panko breadcrumbs until fully coated.
Arrange on the baking sheet and bake for 25-30 minutes until the internal temperature reaches 165 degrees Fahrenheit (74 degrees Celsius).
In a small saucepan, melt butter over medium heat and whisk in flour for 1 minute to create a roux.
Slowly whisk in milk and continue stirring until the sauce thickens and reaches a simmer.
Remove sauce from heat and whisk in Dijon mustard, Parmesan cheese, and chopped parsley.
Remove toothpicks from the baked chicken and serve immediately with the cream sauce.