Pound chicken to 1/4-inch thickness. Add ham and cheese; fold and secure.
Set up flour (with salt, pepper), egg (with water), and breadcrumb (with Parmesan, garlic powder, parsley) stations. Taste and adjust the breadcrumb seasoning.
Dredge chicken in flour, egg, and breadcrumbs.
Cook in butter and oil for 4–5 minutes per side; bake at 375°F for 15–20 minutes.
(Optional) Heat cream, mustard, Parmesan, and pepper for the sauce.
Serve on a white ceramic plate with parsley, sauce, potatoes, and wine.