Butterfly the chicken breasts and pound between plastic wrap to a uniform 1/4-inch thickness using a meat mallet.
Season chicken with salt and pepper, then layer one slice of ham and one slice of Swiss cheese on each breast, leaving a small border at the edges.
Roll the chicken tightly from the narrow end and secure the seams with toothpicks.
Prepare a standard breading station: flour mixed with garlic powder and paprika in the first bowl, beaten eggs in the second, and Panko breadcrumbs in the third.
Dredge each chicken roll in flour, dip into the egg wash, then coat thoroughly in Panko, pressing firmly to ensure adhesion.
Heat vegetable oil in a heavy-bottomed skillet over medium-high heat to 350°F (175°C); sear chicken rolls until golden brown on all sides, roughly 2-3 minutes per side.
Transfer seared chicken to a wire rack set over a baking sheet and bake at 375°F (190°C) until the internal temperature reaches 165°F (74°C).
While chicken bakes, melt butter in a saucepan over medium heat, whisk in flour to create a roux, and cook for 60 seconds without browning.
Slowly incorporate milk while whisking constantly until the sauce thickens and reaches a simmer.
Remove sauce from heat; stir in Dijon mustard and Parmesan cheese until smooth; season with salt and pepper.
Remove toothpicks from the chicken and serve immediately topped with the warm Dijon sauce.