Begin by combining the flour, dried thyme, and freshly ground black pepper in a shallow dish. Dredge each chicken breast in the flour mixture, shaking off any excess flour.
Heat the olive oil and 1 tbsp of unsalted butter in a large skillet over medium-high heat. Use the tongs to add the pieces of chicken. Cook until the chicken is lightly browned on each side, about 2-3 minutes per side. Use the tongs to flip them.Transfer the cooked chicken to a plate and cover to keep warm. Don’t clean the skillet quite yet! You’ll want those brown bits on the bottom. Melt the remaining butter in the same skillet. Add mushrooms and onions and sauté until golden brown and soft, about 2-4 minutes.
Lower heat to medium-low. Pour in the Marsala wine. Let it simmer for about 10 minutes, scraping up the brown bits on the bottom of the skillet. This gets rid of the alcohol in the wine.
Stir in the chicken stock and heavy cream, bring the mixture to a gentle simmer, and cook until the sauce thickens.
Return the chicken to the skillet, spooning the sauce over the chicken, and cook until the chicken is cooked through. Occasionally spoon a little more sauce over the pieces to help with flavoring and to keep the chicken from drying out. When it’s done, an internal thermometer inserted into one of the chicken pieces should read 165 degrees Fahrenheit (74 Celsius).
Arrange on plates with your favorite sides and parsley for garnish. Serve hot.