Transfer the chicken to a warm platter and serve hot.
Heat oil in a large skillet or sauté pan over medium-high heat. Season the herbed, stuffed breasts with salt and pepper.
At this point they can be covered with plastic wrap and stored in the refrigerator for up to 24 hours.
Coat all the breasts with herbs and set them on a plate until ready to cook.
One at a time, put a filled chicken breast in the herb mixture, press to adhere as many herbs as you can, then turn and coat the other side with herbs.
Mix the parsley, rosemary and sage together in a wide shallow bowl or pie plate.
Lay a slice of ham and a slice of cheese horizontally along the bottom half of each breast, fold a 1/2-inch strip of each side inward and then fold the top over to enclose the filling completely.
Season the top of each piece of chicken with salt and pepper
When oil is hot, carefully lower chicken into pan, reduce heat to medium and cook, uncovered, until the underside is a deep brown color, about 5 to 6 minutes. Turn the chicken over and cook on the other side until well browned and the chicken is cooked through, another 5 to 6 minutes.
Pound chicken breasts, one at a time, between sheets of waxed paper or parchment paper with a rubber or wood mallet or the side of an empty wine bottle until they are very thin 1/4-inch thick and about 5 inches wide and 7 inches long