In a ziplock bag, combine flour, ¾ teaspoon salt, and ¼ teaspoon pepper. Add the chicken, seal the bag, and shake to coat evenly.

Heat olive oil and 2 tablespoons butter in a large skillet over medium-high heat. Shake off excess flour and add chicken to the pan. Cook for 5 to 6 minutes, turning once, until golden and just cooked through. Transfer chicken to a plate and set aside.

Melt remaining 1 tablespoon butter in the same pan. Add mushrooms and cook for 3 to 4 minutes, stirring often.

Stir in shallots, garlic, and ¼ teaspoon salt. Cook for 1 to 2 minutes until fragrant.

Pour in chicken broth, Marsala wine, heavy cream, thyme, ¼ teaspoon salt, and ⅛ teaspoon pepper. Scrape up any brown bits from the pan. Bring to a gentle boil and cook uncovered for 10 to 15 minutes, until the sauce reduces by half and thickens slightly.

Return the chicken and juices to the pan. Reduce heat to low and simmer for 2 to 3 minutes, until the sauce thickens slightly more.

Garnish with chopped parsley, if using, and serve hot.