Pound and dredge the chicken: Pat the chicken cutlets dry, then pound to an even 1/2-inch thickness so they cook evenly. On a plate, whisk together the flour, salt, and pepper. Dredge each cutlet in the seasoned flour, shaking off the excess.
Sear the chicken golden: Heat the olive oil and 2 tablespoons butter in a large, heavy skillet over medium-high heat. When the butter foams, add the chicken cutlets in a single layer and sear 3-4 minutes per side until deep golden. Transfer to a plate and keep warm.
Brown the mushrooms and shallots: Reduce the heat to medium. In the same skillet, add the sliced mushrooms in a single layer and let them sit undisturbed for 2-3 minutes to develop a deep brown color. Stir, then add the shallots and cook 2-3 more minutes until soft. Add the garlic and cook 1 minute until fragrant.
Deglaze with Marsala wine: Pour in the Marsala wine, scraping up all the browned bits from the bottom of the pan with a wooden spoon. Let it simmer for 2-3 minutes until reduced by about half. Add the chicken broth and simmer 5 minutes more until the sauce thickens slightly.
Finish the sauce: Reduce the heat to low. Stir in the heavy cream if using, then swirl in the cold butter one piece at a time until the sauce turns glossy and silky. Taste and adjust salt and pepper.
Combine and serve: Return the chicken cutlets to the skillet, spooning the sauce generously over each one. Simmer gently for 2 minutes to warm through and let the chicken absorb the sauce. Serve hot over buttered pasta or mashed potatoes.