Working with 1 chicken cutlet at a time, place between 2 sheets of parchment paper or plastic wrap. Using a meat mallet or rolling pin, pound each cutlet to ¼-inch thickness. Season the chicken all over with 1 teaspoon of the salt and pepper. Place the flour in a shallow bowl or pie pan. Press each cutlet into the flour, and turn to coat evenly. Heat 2 tablespoons of the butter and 2 tablespoons of olive oil in a large skillet over medium-high heat. Once the oil is glistening and the butter begins to bubble, working in batches as needed, place the floured chicken breasts in the skillet. Cook, undisturbed, until golden brown 3 to 4 minutes each side. Transfer the chicken to a plate. Repeat with remaining chicken. Add the remaining 1 tablespoon of olive oil to the skillet. Reduce the heat to medium and add the shallot and mushrooms. Cook, stirring often, until the mushrooms and shallots begin to soften, about 3 to 4 minutes. Stir in the garlic, thyme, and remaining ½ teaspoon salt and cook until fragrant, about 1 more minute. Stir in the stock, Marsala, and heavy cream. Scrape up any brown bits from the bottom of the skillet. Increase the heat to high and bring to a boil, then reduce to medium-low and simmer, stirring occasionally until the wine is reduced by half and the sauce is thick enough to coat the back of a spoon, about 10-12 minutes. Return the chicken to the skillet. Continue to simmer until the internal temperature of the chicken reaches 165°F on an instant read thermometer, about 5 minutes. Transfer the chicken to a serving platter. Stir the remaining 2 tablespoons of butter into the sauce until incorporated. Pour the sauce over the chicken and sprinkle with parsley, if using.