Place the flour on a plate. Season the chicken with ½ teaspoon each salt and pepper. One cutlet at a time, coat the chicken in the flour (tapping off any excess).
Heat the oil in a large skillet over medium-high heat. Working in 2 batches, cook the chicken until golden, 3 to 5 minutes per side; transfer to a plate.
Add the mushrooms and 2 tablespoons water to the skillet. Season with ¼ teaspoon each salt and pepper and cook, tossing occasionally, until the mushrooms are browned and tender, 4 to 6 minutes. Add the Marsala and cook until reduced by half, 3 to 5 minutes.
Add the broth to the skillet and bring to a simmer. Add the butter and swirl until melted. Return the chicken to the skillet, add the parsley, and cook until heated through, 2 to 3 minutes.
Meanwhile, steam the green beans in a steamer basket in a large saucepan until tender, 6 to 8 minutes. Serve with the chicken and mushroom sauce.