Butterfly chicken breasts by slicing them horizontally in half. Cut them into smaller pieces, then trim any membranes or fats.
Place the chicken 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. When it gets very hot, add olive oil to coat the bottom. Sear chicken until lightly browned on each side, about 2 minutes. Transfer to a plate.
Sear the mushrooms, adding oil as needed, for about 2 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 5 minutes. Add chicken stock/broth and let the sauce simmer for another 5 minutes.
Bring the chicken back and add cream.
Spoon the sauce over the chicken, then let it heat through for a minute. Turn off the heat.
Garnish with fresh parsley.