Pat the chicken dry with paper towels. Use a sharp knife to cut each chicken breast in half horizontally to create two chicken breast halves.
Place the chicken breast halves between two sheets of plastic wrap and use a meat mallet or rolling pin to pound them into thin and evenly (be careful not to pound so hard that the meat tears.).
Top each pieces of chicken with a slice of ham and a ham handful of shredded cheese. Roll tightly, tucking the sides a little, and place on a new piece of plastic wrap.
Wrap the chicken tightly in the plastic wrap, pinching the excess plastic on the sides to create a tootsie-roll shape and twisting them to create a firm chicken roll. Refrigerate the wrapped chicken bundles for at least 30-minutes or up to one day in advance.
Preheat oven to 400 degrees F. Remove chicken from fridge, unwrap and season with salt and pepper. Add cornflakes to a food processor and pulse into fine crumbs. Add crumbs to a shallow dish or pie plate. Melt butter in a separate shallow dish.
Dip the chicken bundles in melted butter, and then into the cornflake crumbs, pressing lightly to help the crumbs stick to the chicken. Transfer to a lightly greased baking sheet.
Bake on the center oven rack for about 30 minutes, or until chicken is cooked through (160 degrees on a thermometer inserted into the chicken, not the filling.)
Make the sauce by stirring the mayo and mustard together. Taste and add more mustard, as needed, to taste.
Sauce: I use a very simple sauce for my chicken cordon bleu; mayonnaise and mustard mixed together. This is how I ate it growing up and I think it's delicious, but here is a more traditional chicken cordon bleu sauce recipe if you prefer.Make ahead instructions: Prepare the recipe through step 4 (wrapping stuffed chicken breasts and placing them in the refrigerator). Store for up to one day and prepare as directed.Freezing instructions: Prepare the recipe through step 4 and place wrapped chicken breasts in a freezer safe container or ziplock bag. Freeze for up to 3 months. Thaw in the refrigerator overnight and continue with step 5 of the recipe.