Season the chicken breasts with salt and pepper, then dredge them in flour, shaking off the excess.
Heat olive oil in a large skillet over medium-high heat. Add chicken breasts and cook until golden-brown on both sides. Remove and set aside.
In the same skillet, add butter and sauté mushrooms until browned. Add garlic and cook until fragrant.
Pour in Marsala wine and chicken broth. Stir and bring to a boil.
Return chicken to the skillet. Simmer until chicken is cooked through and the sauce has reduced, about 15 minutes.
Meanwhile, blanch the green beans in boiling water until tender-crisp, then drain.
Serve the chicken topped with sauce and mushrooms alongside the green beans.