**Prepare the Chicken:** Place the chicken breasts between two sheets of plastic wrap and pound them to an even thickness. Season with salt and pepper, then dredge each piece lightly in flour, shaking off the excess.
**Sear the Chicken:** Heat olive oil and 1 tbsp of butter in a large skillet over medium-high heat. Add the chicken breasts and cook for 4-5 minutes per side until golden brown and cooked through. Remove from the skillet and set aside.
**Cook the Mushrooms:** In the same skillet, melt another tablespoon of butter. Add the sliced mushrooms and sauté for 5-7 minutes until they are golden and softened. Stir in the minced garlic and cook for another minute until fragrant.
**Deglaze the Pan:** Pour in the Marsala wine and chicken broth, scraping up any browned bits from the bottom of the skillet. Bring to a simmer and let it reduce slightly, about 3-4 minutes.
**Add the Cream:** Stir in the heavy cream and thyme or Italian seasoning. Allow the sauce to simmer for 3-5 minutes until it thickens slightly.
**Combine:** Return the chicken to the skillet, spooning the sauce over it to coat. Simmer for another 2-3 minutes to warm the chicken through.
**Serve:** Garnish with fresh parsley and serve hot over pasta, mashed potatoes, or rice.