Dredge the chicken. Whisk together the flour, salt, and pepper in a shallow dish. Then, coat each chicken cutlet evenly on both sides with the flour mixture, shaking off any excess. Set aside.
Cook the chicken. Heat 1 tablespoon of olive oil in a large frying pan over medium heat. Once hot, add the chicken cutlets in a single layer (working in batches if necessary to avoid overcrowding). Cook for 5-6 minutes on each side until golden brown. The chicken does not need to be fully cooked through at this point. Then, transfer to a plate and set aside.
Cook the veggies. In the same pan, melt 1 tablespoon of butter over medium heat. Then, add the sliced mushrooms and sauté for 6-8 minutes until they release moisture and turn golden brown. Next, stir in the diced onion and garlic, cooking for another 5 minutes until softened and fragrant.
Add wine and simmer. Pour in the Marsala wine and simmer over medium heat. Allow the wine to reduce by half, taking about 3-4 minutes. Stir occasionally to prevent sticking.
Thicken the sauce. Once the wine has reduced, stir in the chicken stock, heavy cream, and thyme. Continue simmering the sauce for about 3 minutes, until it thickens slightly and coats the back of a spoon.
Add chicken. Return the chicken cutlets and any juices that have accumulated on the plate into the pan. Stir gently to coat the chicken with the sauce. Let the chicken simmer in the sauce for another 5-7 minutes, or until the sauce has reduced by half and the chicken is cooked through.
Garnish and serve. Stir in the fresh chopped parsley. Taste and adjust the seasoning with salt and pepper if necessary. Serve hot over pasta, mashed potatoes, or rice.