Place the chicken cutlets between 2 pieces of plastic wrap and pound with a meat mallet (or the flat side of a chef’s knife) until about 1/3 inch thick. Sprinkle with 1/4 teaspoon salt and 1/4 teaspoon pepper.
Put the flour on a medium plate. Heat the oil in a large nonstick skillet over medium-high heat. Dredge the chicken in the flour to fully coat, shaking off any excess. Add the chicken to the skillet and fry until fully cooked and golden brown, about 4 minutes per side. Transfer to a platter and tent with foil to keep warm.
Add 1/2 cup of the broth, the sun-dried tomatoes and rosemary to any remaining drippings in the skillet and cook, stirring frequently, for 1 minute to plump the tomatoes. Add the mushrooms, 1/4 teaspoon salt and 1/2 teaspoon pepper and cook until the mushrooms are soft, about 5 minutes. Add the marsala and bring to a boil. Add the remaining 1/4 cup broth and the butter and simmer until the butter is fully melted, about 30 seconds.
Spoon the mushroom mixture and sauce over the chicken, sprinkle with the parsley and serve.