Preheat oven to 375°F (190°C) and line a baking sheet with parchment paper.

Place chicken breasts between sheets of plastic wrap and pound to a uniform 1/4-inch thickness using a meat mallet.

Season chicken with salt and pepper; place one slice of ham and one slice of Swiss cheese on each breast, roll tightly, and secure with toothpicks.

Set up a dredging station with three shallow bowls containing flour, beaten eggs, and Panko breadcrumbs respectively.

Coat each chicken roll in flour, dip in egg wash, and press into Panko breadcrumbs until fully coated.

Arrange chicken on the baking sheet, drizzle with 2 tablespoons melted butter, and bake for 20-25 minutes until the internal temperature reaches 165°F (74°C).

Prepare the sauce by melting 2 tablespoons butter in a saucepan over medium heat, then whisking in 2 tablespoons flour for 60 seconds to create a roux.

Gradually whisk in milk and continue stirring until the liquid thickens and reaches a light boil.

Reduce heat to low and stir in shredded Swiss, Parmesan, and Dijon mustard until the cheese is completely melted and the sauce is smooth.

Remove toothpicks from the rested chicken rolls and serve immediately topped with the warm cheese sauce.