Prepare the chicken cutlets. If your chicken breasts are thick, slice them in half horizontally. Place the cutlets between two pieces of plastic wrap and pound them to an even 1/4-inch thickness using a meat mallet.

In a shallow dish, whisk together the all-purpose flour, salt, and pepper. Lightly dredge each chicken cutlet in the flour mixture, shaking off any excess.

Heat the olive oil and 2 tablespoons of butter in a large skillet over medium-high heat. Once the butter is melted and sizzling, add the chicken cutlets in a single layer (work in batches if needed). Sear for 3-4 minutes per side until golden brown. Remove the chicken from the skillet and set aside on a plate.

Reduce the heat to medium and add the remaining 2 tablespoons of butter to the skillet. Add the sliced mushrooms and cook, stirring occasionally, until they have browned and released their moisture, about 5-7 minutes.

Add the minced shallot and garlic to the skillet and sauté for 1 minute until fragrant.

Pour in the dry Marsala wine to deglaze the pan. Use a wooden spoon to scrape up any browned bits stuck to the bottom of the skillet. Allow the wine to simmer and reduce by about half, which should take 2-3 minutes.

Stir in the chicken broth and heavy cream. Bring the sauce to a gentle simmer and let it cook for 5-7 minutes, stirring occasionally, until it begins to thicken.

Return the seared chicken cutlets and any accumulated juices back to the skillet. Spoon some of the sauce over the chicken. Let everything simmer together for about 5 more minutes, until the chicken is cooked through and tender.

Garnish with fresh chopped parsley and serve immediately.