Once all of the chicken has been cooked, use the same skillet and add the remaining butter and olive oil. Add the garlic and shallots and cook until soft and golden, ~2 minutes. Add the mushrooms, season with a small pinch of salt and pepper. Cook, stirring occasionally, until golden, ~5 minutes. Sprinkle in the 1 tsp of flour and cook, stirring, for ~30 seconds. Add the Marsala wine, chicken broth and parsley.
Cook, stirring and scraping up any browned bits from the bottom of the pan with a wooden spoon, until thickened, ~2 minutes.
Return the chicken to the skillet with the mushrooms, reduce heat to low, cover, and simmer in the sauce to let the flavors blend, ~4-5 minutes.
To serve, put a piece of chicken on each of 4 serving plates. Spoon the mushrooms and sauce evenly over top and serve hot.