Slice chicken breasts. Cut the chicken breasts in half lengthwise, making 4 thinner cutlets.
Season the chicken. Mx the flour, salt, and pepper in a shallow dish or bowl. Dredge the chicken in the mixture, coating both sides.
Sear the chicken. Heat the oil and butter in a large skillet over medium-high heat. Sear the chicken for 3 to 4 minutes per side, until it has a golden crust. It's okay if it's not fully cooked through at this point, as it will simmer again. Use tongs to remove the chicken to a plate.
Saute the mushrooms. Melt the butter in the pan. Add the mushrooms and saute for 2 to 3 minutes, stirring occasionally, until lightly browned. Add the garlic and shallot and saute for another 30 seconds.
Make the sauce. Add the broth, Marsala wine, and heavy cream to deglaze the pan. Stir together and bring the liquid to a boil, then lower the heat and simmer until it's thickened and reduced by half, about 10 to 15 minutes.
Return the chicken. Add the chicken back to the pan, spoon the sauce over it, and simmer until the chicken is warmed through, about 2 to 3 minutes. Sprinkle with freshly chopped parsley before serving.