Slice the chicken breasts in half horizontally to create four thinner cutlets. Place them between two pieces of plastic wrap and use a meat mallet or rolling pin to pound them to an even 1/4-inch thickness. Season both sides generously with salt and pepper.

In a shallow dish, combine the all-purpose flour with a pinch of salt and pepper. Lightly coat each chicken cutlet in the flour mixture, shaking off any excess.

In a large skillet over medium-high heat, melt the butter with the olive oil. Place the floured chicken cutlets in the skillet (work in batches if necessary). Cook for 3-4 minutes per side, until golden brown and cooked through. Remove the chicken from the skillet and set it aside.

To the same skillet, add the sliced mushrooms. Cook until they are nicely browned, about 5-7 minutes. Add the minced shallot and garlic, and cook for another minute until fragrant.

Pour in the dry Marsala wine to deglaze the pan, scraping up any browned bits from the bottom. Allow the wine to simmer and reduce by about half. Stir in the chicken broth and heavy cream.

Bring the sauce to a simmer and cook for about 5 minutes until it starts to thicken. Return the cooked chicken to the skillet, spooning sauce over the top. Let it simmer for 1-2 minutes to heat through. Garnish with fresh parsley before serving.