Place chicken breasts between two sheets of plastic wrap and pound with a meat mallet to an even 0.25-inch thickness.

Season both sides of the chicken with salt and black pepper.

Place one slice of ham and one slice of Swiss cheese on each chicken breast, leaving a 0.5-inch border at the edges.

Tuck in the sides and roll the chicken tightly; secure with wooden toothpicks if necessary.

Prepare three shallow bowls: one with flour mixed with garlic powder, one with beaten eggs, and one with Panko breadcrumbs.

Dredge each roll in the flour, dip into the egg wash, and coat thoroughly with Panko breadcrumbs, pressing firmly to adhere.

Heat the butter and vegetable oil in a large skillet over medium-high heat until the butter is foamy.

Add the chicken rolls to the skillet and cook for 4 to 5 minutes per side, turning carefully, until the exterior is golden brown and the internal temperature reaches 165 degrees Fahrenheit.

Remove from heat and allow the chicken to rest for 5 minutes before removing toothpicks and serving.