Preheat oven to 350°F. Pour olive oil into an 8 x 8-inch baking dish; set aside.
Cut each chicken breast in half lengthwise. Place each half of a chicken breast between 2 sheets of plastic wrap; pound with a meat mallet to ¼-inch thickness.
Place egg and water in a shallow dish; beat with a fork until well combined.
In a separate shallow dish, combine breadcrumbs and Parmesan cheese.
Season the chicken with salt and pepper to taste. Place a slice of ham in the center of each chicken breast. Top with a slice of Swiss cheese. Depending on the size of your ham and cheese slices, you may need to trim them or cut them in half so that they are not hanging over the edges of the chicken.
Roll up each chicken breast to enclose the ham and cheese. Secure with wooden toothpicks.
Dip each piece of chicken in the egg mixture, shaking off any excess. Roll the chicken in the breadcrumb mixture, pressing to coat completely.
Place the baking dish in the oven for 2 minutes, or until the oil gets hot. Place the chicken, seam-side down, in the hot dish. Top each piece of chicken with a pat of butter.
Bake, uncovered, for 35 minutes, or until a meat thermometer registers 165°F. Remove the toothpicks before serving; garnish with chopped fresh parsley.