Butterfly chicken breasts by slicing them horizontally in half. You can cut them into smaller pieces, then trim any membranes or fats.
Place the chicken in between sheets of parchment paper or plastic wrap. Pound them thin (about ½ inch) for quick and even cooking. Season them with salt and pepper, then lightly coat them with flour.
Set a large skillet over medium-high heat. Add oil to coat the bottom and let it get very hot. Sear the chicken until lightly browned on each side, about two minutes. Transfer to a plate.
Brown the mushrooms in a bit of oil. Allow them to brown for about two minutes without touching them. Add salt and pepper, then mix them.
Add shallots, garlic, and thyme. Let them cook for a minute.
Deglaze the pan with Marsala wine and scrape the fond or browned bits on the bottom. Let it reduce by half for about five minutes. Add chicken stock and let the sauce simmer for another five minutes.
Bring the chicken back and add cream.
Spoon the sauce over the chicken, then let it heat for a minute. Turn off the heat.
Garnish with fresh parsley.