Slice each chicken breast in half horizontally to create 4 thin cutlets. Season both sides with salt and pepper, then dredge lightly in flour, shaking off excess.
In a large skillet over medium-high heat, add olive oil and 1 tablespoon of butter. Sear the chicken on both sides for about 3–4 minutes per side, or until golden and cooked through. Transfer to a plate and set aside.
In the same pan, melt the remaining butter. Add sliced mushrooms and cook for 5–6 minutes, stirring occasionally, until browned. Stir in the minced garlic and cook for 30 seconds.
Pour in the Marsala wine and scrape up any browned bits from the bottom of the pan. Let the wine simmer for 2–3 minutes to reduce slightly.
Add chicken broth and heavy cream to the pan. Stir and bring to a simmer. Let the sauce cook for another 4–5 minutes, or until slightly thickened.
Place the cooked chicken back into the skillet, spooning the sauce over the top. Simmer for 2–3 minutes until heated through.
Garnish with chopped parsley if desired. Serve hot over mashed potatoes, pasta, or rice.