Preheat oven to 375°F. Line your baking sheet and set up a dredging station: flour in one bowl, beaten eggs mixed with milk in the second, and crumbs plus Parmesan, salt, pepper, and garlic powder in the third.

Place a butterflied chicken breast between plastic wrap and gently pound until about 1/4 inch thick. Season lightly with 1/4 tsp salt and a pinch of pepper.

Spread 1/2 tbsp Dijon mustard over the pounded chicken. Lay one slice of ham and one slice of Swiss cheese on top, leaving a small border for sealing.

Tightly roll the chicken around the filling, tucking in ends. Secure with toothpicks or kitchen twine so the roll holds in the oven.

Dust the roll in flour, dip into the egg wash, then press into the breadcrumb-Parmesan mixture until evenly coated. Repeat with remaining breasts.

Heat olive oil and 1 tbsp butter in an ovenproof skillet over medium-high heat. Sear rolls for 1-2 minutes per side until golden. This step is optional.

Transfer to the preheated 375°F oven and bake for 25-30 minutes, or until an instant-read thermometer reads 165°F in the thickest part. Remove toothpicks before serving.