Season the chicken cutlets on both sides with salt and pepper.
Add the flour to a shallow bowl; dredge the chicken in the flour to coat, shaking gently to remove any excess.
In a 12 inch skillet, melt 2 Tablespoons butter and 2 Tablespoons olive oil over medium-high heat. Add the chicken and sear until browned, about 3-4 minutes on per side. Remove to a plate and cover with foil.
Wipe out the skillet, then heat the remaining 2 Tablespoons butter and 2 Tablespoons olive oil over medium heat until butter is melted.
Add the mushrooms and shallot; sauté over medium-high heat, stirring regularly, about 8 minutes or until mushrooms are well browned.
Add minced garlic and sauté 30 seconds, until fragrant.
Add the marsala wine, chicken broth, 1/2 teaspoon salt and 1/4 teaspoon pepper. Bring to a boil, then reduce heat to medium-low and boil gently for 10-15 minutes, until sauce is thickened and reduced by about half.
Reduce heat to low, then stir in the heavy cream.
Add the chicken back into the skillet and spoon the sauce over it. Cook 3-5 minutes more.
Garnish with fresh parsley as desired.