Place the chicken cutlets between 2 pieces of plastic wrap, parchment paper, or in a gallon freezer bag, in a single layer.
Using a meat mallet or rolling pin, pound the chicken to an even thickness, about ¼-inch thick, being careful not to smash the chicken.
Pat the chicken dry and season all over with salt and pepper.
Coat the chicken with the flour and shake off excess flour.
Heat the oil and 2 tablespoons of the butter in a large skillet over medium-high heat.
Working in batches to prevent crowding, add the chicken to the pan and cook until golden brown on both sides.
Transfer the chicken to a plate (the chicken does not need to be cooked through; it will continue cooking in the sauce).
Add the remaining tablespoon of butter to the pan and set the pan over medium-high heat.
When the butter has melted, add the mushrooms and cook until soft and browned (using a large skillet will prevent the mushrooms from crowding and they will brown better).
Add the shallot and garlic cook for 3 minutes, until soft.
Season the mushrooms with salt and pepper.
Add the broth and Marsala wine to the pan and bring to a simmer, scraping up any browned bits from the bottom of the pan.
Reduce the heat to medium and add the cream.
Simmer, uncovered, until the sauce thickens, darkens, and reduces by about half, about 6 to 8 minutes.
Return the chicken to the pan with any accumulated juices on the plate.
Simmer until the chicken is cooked through and the sauce thickens slightly more, about 3 minutes.
Garnish with parsley and serve.