Place each chicken breast between two pieces of plastic wrap and gently pound to an even thickness of about ½ inch using a meat mallet or rolling pin.
In a shallow dish, mix the flour with salt and pepper. Dredge each chicken breast in the flour, shaking off any excess.
Heat 2 tablespoons of olive oil and 2 tablespoons of butter in a large skillet over medium-high heat. Add the chicken breasts and cook for about 4-5 minutes on each side, until golden brown and cooked through. Remove the chicken from the skillet and set aside on a plate.
In the same skillet, add the remaining olive oil and butter. Add the sliced mushrooms and sauté for about 5-7 minutes until they are browned.
Add the minced garlic and sauté for an additional minute until fragrant.
Pour in the Marsala wine, scraping up any browned bits from the bottom of the skillet. Allow it to simmer for about 2-3 minutes.
Add the chicken broth and return the chicken to the skillet. Simmer for another 10 minutes to allow the flavors to meld and the sauce to thicken slightly.
Taste and adjust seasoning with salt and pepper if necessary.
Serve the chicken hot, drizzled with the sauce, and garnish with chopped parsley.