Lay each chicken breast between sheets of plastic wrap and pound with the flat side of a meat mallet until ¼-inch thick. Season with salt and pepper and set aside.
Heat the olive oil and butter in a 12-inch skillet over medium-high heat. Dredge the chicken breasts in flour and shake off the excess. Fry the chicken breasts until golden, about 5 minutes on each side. Transfer the chicken to a platter and keep warm.
Reduce heat to medium and sauté the shallots until they begin to turn golden, about 1 minute. Add the mushrooms and sauté until their moisture has evaporated and they are browned, about 6 to 7 minutes. Season lightly with salt and pepper. Increase the heat to medium-high and add the Marsala wine. Boil for about 15 to 30 seconds, scraping the pan to loosen the brown bits. Add the chicken broth and milk and continue to cook over medium heat until the sauce thickens slightly, about 6 minutes. Add the browned chicken breasts, cooking just until warmed, about 1 minute. Serve each chicken breast with the mushroom sauce spooned over it.