Begin by pounding the chicken breasts to an even thickness, about 1/2 inch, using a meat mallet. This helps the chicken cook evenly. Season the chicken breasts with salt and pepper on both sides. Dredge the chicken in the flour, shaking off any excess.
In a large skillet, heat 1 tablespoon of butter and 1 tablespoon of olive oil over medium-high heat. Once the butter is melted and the oil is hot, add the chicken breasts. Cook for about 3-4 minutes on each side, or until they are golden brown and cooked through. Remove the chicken from the skillet and set aside.
In the same skillet, add the remaining butter and olive oil. Add the sliced mushrooms and minced shallots and sauté until both are tender and mushrooms are golden brown. Add the minced garlic and sauté for about 1 minute until fragrant, being careful not to burn it. Pour in the Marsala wine and chicken broth, and bring the mixture to a simmer. Scrape the bottom of the skillet to incorporate any flavorful bits from the chicken.
Let the Marsala wine and broth mixture simmer for about 5-7 minutes, allowing it to reduce slightly. Pour in the heavy cream and continue to simmer for another 3-4 minutes until the sauce has thickened.
Return the cooked chicken breasts to the skillet, nestling them into the sauce. Gently simmer for an additional 2-3 minutes to heat the chicken through.
Sprinkle the chopped fresh parsley over the Chicken Marsala for a burst of color and flavor. Taste and adjust the seasoning with salt and pepper if needed.
Serve the Chicken Marsala over cooked pasta, rice, or mashed potatoes. Spoon the flavorful Marsala sauce and mushrooms over the chicken. Garnish with additional chopped parsley if desired.