Slice chicken horizontally into cutlets and pound between sheets of parchment to an even 1/2-inch thickness. Divide into four equal portions. Sprinkle with salt and pepper, then coat lightly with flour, shaking excess back into the dish.

Heat oil in a large skillet over medium-high heat. Brown chicken in two batches, 3–4 minutes per side. Move to a plate; check its temperature later rather than assuming the browning has cooked the centers.

Lower heat to medium. Melt butter in the skillet and cook mushrooms 6–8 minutes, stirring until their liquid evaporates and edges brown. Add shallot, garlic and thyme; stir for 1 minute.

Take the pan away from the heat before pouring in Marsala. Return to medium heat, add broth and scrape up the browned bits. Simmer uncovered 5–7 minutes to reduce. Stir in cream and simmer gently 2–3 minutes until lightly thickened.

Return chicken and all plate juices. Simmer gently 2–5 minutes until several thick spots reach 165°F and the sauce coats the cutlets. If needed, remove cooked chicken while the sauce reduces a little longer. Divide all chicken, mushrooms and sauce among four plates.