Season the chicken with salt and pepper and dust both sides lightly with the flour.
Heat the oil in a wide skillet over medium-high and cook the chicken for 3 to 4 minutes per side until browned, then transfer it to a plate.
Add the mushrooms and cook undisturbed for 3 minutes, then stir and cook for 3 minutes more until browned.
Pour in the Marsala and broth and simmer for 6 to 8 minutes until reduced by about half.
Return the chicken to the pan until it reaches 165°F, remove from the heat, and swirl in the cold butter until the sauce looks glossy.