Crush or grind the Cheerios finely. Tip - use a food processor.
Cut each chicken breast in half lengthwise into 2 equal sized pieces for a total of 4.
Pound them to about 1/4 inch thick (but be careful not to tear the meat)
Season chicken with salt and pepper.
Top each piece of chicken with a slice and a half of swiss cheese, followed by one or two slices of Ham.
Using 3 dishes (one for flour, one for the eggs, and one for the crushed Cheerios), roll the chicken in flour, then coat the chicken in the beaten egg, covering completely. Finally, coat all sides of the chicken in the crushed Cheerios.
Let the chicken sit in the fridge for at 45 minutes (optional).
In a skillet over medium heat, melt some butter, and once it's nice and bubbly, place each piece of breaded chicken in the butter.
Cook each side of the chicken for about one minute to get a nice brown crust.
Bake in the oven for 20-25 minutes or until done.