In food processor, combine butter, mustard, garlic, shallot, lemon juice, salt, pepper and herbs, and pulse into a smooth compound. Transfer to a baggie, roll into a log and chill a bit.
Pre-heat oven to 275°F.
Butterfly chicken breasts across but do not cut all the way through. Open each piece like a book and cover with parchment; pound very thin. Season each cutlet with salt and pepper. Arrange a slice of ham and cheese on each breast, top with a slice of herb butter and flap chicken over to form a stuffed cutlet.
Coat cutlets in flour, egg and breadcrumbs mixed with grated cheese.
Heat an even layer of olive or vegetable oil, about 1/8 inch, over medium to medium-high heat. Add chicken and cook 4 minutes or so on each side to deeply golden.
Keep chicken warm in oven on wire rack over baking sheet.
Serve with egg noodles or mashed potatoes dressed with half of the remaining herb butter. Add peas to egg noodles or mix into mashed potatoes. If serving with mashed potatoes, mash with milk or stock to desired consistency.