Season both sides of the chicken breasts with salt and pepper. Place the flour in a shallow dish and dredge the chicken pieces, shaking off any excess.
In a large skillet, heat the olive oil over medium-high heat. Add the chicken and cook until golden brown and cooked through, about 3–4 minutes per side. Transfer to a plate and keep warm.
Add the mushrooms to the same skillet and sauté until they release their moisture and start to brown, about 5–6 minutes. Stir in the garlic for about 30 seconds until fragrant.
Pour in the Marsala wine, using a wooden spoon to scrape up any browned bits from the bottom of the pan. Simmer until the liquid reduces by about half, 3–4 minutes.
Add the chicken broth and bring to a gentle simmer. Stir in the cream and butter, and cook until the sauce thickens to a glossy consistency, about 2–3 minutes. Return the chicken to the pan and simmer briefly to heat through and meld the flavors.
Season with thyme, salt, and pepper to taste. If the sauce is too thin, simmer a bit longer; if too thick, loosen with a splash of broth. Finish with a final pat of butter for extra gloss if desired.
Garnish with chopped parsley and serve hot over mashed potatoes, pasta, or a bed of sautéed greens to soak up every bit of sauce.