Prepare the Chicken Coating: In a large resealable plastic bag, combine the flour with half of the salt (about ⅜ teaspoon) and half of the pepper (about ⅛ teaspoon). Add the chicken pieces, seal the bag tightly, and shake well to coat each piece thoroughly with the seasoned flour. Set aside while you heat the pan.
Sear the Chicken Cutlets: Heat the olive oil and 2 tablespoons of butter in a large skillet over medium-high heat. Once hot, add the floured chicken cutlets in a single layer, being careful not to crowd the pan. Cook for about 2.5 to 3 minutes per side, until the chicken is beautifully golden brown and just cooked through. Remove the chicken to a plate and cover loosely with foil to keep warm.
Sauté the Mushrooms and Aromatics: Lower the heat to medium and add the remaining tablespoon of butter to the skillet. Add the sliced mushrooms and cook, stirring often, until they begin to brown and release their moisture, about 3 to 4 minutes. Stir in the minced shallots, garlic, and the remaining salt and pepper. Cook for an additional 1 to 2 minutes until fragrant.
Deglaze and Simmer the Sauce: Pour in the chicken broth, Marsala wine, and heavy cream. Add the chopped thyme. Use a wooden spoon to scrape up any browned bits stuck to the bottom of the pan — these will add fantastic depth to the sauce. Bring the mixture to a gentle boil, then reduce the heat to medium-low and let it simmer uncovered for about 10 to 15 minutes. The sauce should reduce by half, becoming slightly thicker and darker, but still pourable.
Finish and Serve: Return the cooked chicken cutlets and any juices that collected on the plate back to the pan. Simmer gently on low heat for 2 to 3 minutes, allowing the chicken to warm through and the sauce to thicken a bit more. Taste and adjust seasoning if needed. Sprinkle with fresh parsley if desired and serve immediately.