Whisk together flour with salt and pepper in a shallow bowl.
Next, whisk together an egg with milk, salt, and pepper in a separate shallow bowl.
In another shallow bowl whisk together breadcrumbs with salt and pepper and set aside.
Slice the chicken breasts in half widthwise.
Pound the chicken breast out gently until it is has essentially doubled in size or is about a ¼ to ⅓” thick.
Season the chicken with salt and pepper.
Layer on a slice of ham and then 3 half slices of Swiss cheese.
Fold in the sides slightly and then roll the chicken forward making sure the ingredients stay inside and are enclosed inside. Repeat until all the chicken is stuffed.
Lightly coat the chicken roll in the flour and pat off any excess and then dredge in the egg wash on all sides.
Coat completely in the breadcrumbs and repeat until all the chicken is coated.
Fry the chicken cordon bleu minutes in about a ¼ cup of neutral flavored oil in a non-stick skillet over medium heat for 3 to 4 on all sides or until golden brown and cooked throughout when reading 165° F internally. See chef notes.
Remove the chicken and discard all but 2 tablespoons of the frying oil.
Whisk in the flour until combined and then pour in the chicken stock.
Cook on high heat until thick and then finish with butter, salt, and pepper.
Slice and serve with the simple pan gravy.
