Add the flour to a shallow plate and season with salt and pepper. Coat the chicken pieces with the flour and tap off any excess.
Set a 12-inch cast-iron pan over medium heat, add enough oil to coat the bottom of the pan and heat until hot. Add the chicken in batches and cook until browned, 2 to 3 minutes. Flip and cook on the other side until browned. Keep warm, covered, on a serving plate.
Add about 1 tablespoon more oil to the pan, add the garlic, shallot and mushrooms and saute for about 1 minute. Add the prosciutto, thyme and half of the parsley and cook 1 more minute. Add the Marsala to the pan and deglaze, scraping up any browned bits from the bottom. Cook until the wine is reduced by about three-quarters, 1 to 2 minutes. Add the chicken stock and cook until the sauce is heated through and thickened, 4 to 5 minutes. Remove from the heat and stir in the butter one piece at a time. Spoon the sauce over the chicken and garnish with the remaining parsley. Serve with warm buttered fettuccine.
If the sauce seems too thin, rub about a tablespoon of butter with about 2 tablespoons of the seasoned flour to a crumby texture and whisk it into the sauce a bit at a time. Simmer until thickened.