Lightly pound out cutlets to ¼" thick. Season with a pinch each of salt and pepper.
Place flour in in a shallow dish and lightly dredge the chicken cutlets in the flour shaking off the excess.
Heat a large non-stick skillet over medium-high heat. Add the butter and olive oil to the pan. Swirl until butter has melted. Add the chicken, in batches to prevent overcrowding, and cook until slightly golden on both sides, ~3 minutes per side. Transfer to a baking dish and keep warm in the oven.
Sauce
Once all of the chicken has been cooked, use the same skillet and add the remaining butter and olive oil. Add the garlic and shallots and cook until soft and golden, ~2 minutes. Add the mushrooms, season with a small pinch of salt and pepper. Cook, stirring occasionally, until golden, ~5 minutes. Sprinkle in the 1 tsp of flour and cook, stirring, for ~30 seconds. Add the Marsala wine, chicken broth and parsley.
Cook, stirring and scraping up any browned bits from the bottom of the pan with a wooden spoon, until thickened, ~2 minutes.
Return the chicken to the skillet with the mushrooms, reduce heat to low, cover, and simmer in the sauce to let the flavors blend, ~4-5 minutes.
To serve, put a piece of chicken on each of 4 serving plates. Spoon the mushrooms and sauce evenly over top and serve hot.