Place each chicken breast between two pieces of plastic wrap on a cutting board. Using the flat side of a meat mallet (not the sharp pointed side), or a rolling pin, pound the breasts to about 1/2 inch thick.
In a large skillet, add the tablespoon of olive oil and heat to medium heat.
Add prosciutto ribbons and cook until crispy, stirring constantly. Remove from pan when brown and fat has rendered. Reserve crispy prosciutto in a small dish.
Put the flour in a shallow plate and season with the salt and pepper. Mix the seasonings in with a fork to evenly distribute.
In the same skillet used to cook the prosciutto, heat the oil and butter over medium-high heat. When the oil is hot, dredge the chicken breasts in the seasoned flour. Shake off the excess.
Slip the chicken breasts into the pan and fry for 5 to 7 minutes on each side, until golden. You can do this in batches if all the chicken doesn't fit comfortably in the one pan. Remove the chicken to a large platter in a single layer, and cover with foil to keep warm while preparing the pan sauce.
Lower the heat to medium and add the mushrooms to the remaining oils in the pan. If there is not enough oil, add another drizzle of olive oil to coat the mushrooms. Saute the mushrooms until they are nicely browned and their moisture has evaporated.
During the last minute of cooking the mushrooms, add the minced garlic to the pan and cook for 30-60 seconds.
Pour the Marsala wine into the pan, deglazing the pan and scrapping up any browned bits. Let the Marsala come to a boil for a few seconds to release the alcohol.
Add the chicken stock and simmer for a few minutes to reduce the sauce slightly.
Stir in the remaining butter and return the chicken breasts to the pan. Simmer the chicken gently to heat the chicken through. Taste and season with salt and pepper as necessary.
Garnish with chopped parsley and reserved crisped Prosciutto before serving.