Preheat oven to 350 degrees F
Place the chicken breasts on a cutting board lined with plastic wrap. Cover the chicken with another sheet of plastic wrap and using a meat hammer lightly pound the chicken breasts (see notes).
Turn the chicken so the shiny side is down, and place two slices of the ham in the middle of the breast.
Add half of the shredded gruyere to the center of the ham and fold the ham over to make a packet with the cheese in the middle of the ham.
Wrap the chicken breast around the ham packet, tucking all the ends in to seal the cordon bleu as much as possible.If you have time refrigerate the stuffed chicken for 20 minutes, it will help it keep its form. Set up a breading station with flour seasoned with sea salt and black pepper, egg wash, and panko bread crumbs.
Coat the stuffed chicken breasts with the seasoned flour, using the opportunity to form the formed the stuffed breast into an oblong or round shape.
Place the floured chicken into the egg wash coating completely. Use one whole egg whisked with cold water or milk to make the egg wash. Coat the egged chicken with the bread crumbs. Make sure you get a good coating of each ingredient on the chicken breasts, completely coating them and making sure the stuffing is completely covered by the chicken.If you see any spots that aren't well coated, dip that spot in the eggwash and apply more breadcrumbs.You can hold the breaded chicken breasts in the refrigerator for up to 8 hours. After holding you may need to recoat them in breadcrumbs if the chicken appears wet. Heat a large saute pan on medium-high heat and add about half an inch of vegetable oil. When the oil begins to shimmer carefully place the cordon bleu into the oil using tongs, seam side down.
Allow to cook for about 2 minutes, then turn to the other side. Continue this process until all the sides of the cordon bleu have been browned.
Place the browned cordon bleu on a baking sheet lined with parchment. Place the chicken on the middle rack of the preheated oven and bake for 25-30 minutes or until the chicken has reached an internal temperature of 165 degrees F.