Sprinkle the chicken cutlets with salt and pepper and set aside.
Melt 1 tablespoon of butter and 1 tablespoon of oil in a heavy large skillet over medium-high heat. Add in chicken cutlets 4 at a time (work in batches) and cook until golden brown, about 3 minutes per side. Transfer cooked chicken to a plate and repeat with second batch.
Add 1 tablespoon of oil to the skillet. Add the shallot and garlic. Saute until fragrant, about 30 seconds.
Add a tablespoon of the olive oil, if necessary. Add the mushrooms and saute until tender and the juices evaporate, about 3 minutes. Season with salt.
Add the Marsala. Simmer until the Marsala reduces by half, about 2 minutes.
Add the broth and the rosemary sprigs. Simmer until reduced by half, about 4 minutes. Return chicken to the skillet. Pour in all of the pan juices. Cook just until heated through, turning to coat, about 1 minute. Stir the remaining 1 tablespoon of butter into the sauce. Season the sauce with salt and pepper, to taste.