Place the flour, ¾ teaspoon salt, and ¼ teaspoon pepper in a ziplock bag. Add the chicken to the bag; seal tightly and shake to coat chicken evenly. Set aside.

Heat the oil and 2 tablespoons of the butter in a large skillet over medium-high heat. Add the flour-dusted chicken to the pan and cook, turning once, until golden and barely cooked through, about 5 to 6 minutes total. Transfer the chicken to a plate and set aside.

Melt the remaining tablespoon of butter in the pan. Add the mushrooms and cook, stirring frequently, until browned, about 3 to 4 minutes. Add the shallots, garlic, and ¼ teaspoon salt; cook for 1 to 2 minutes more.

Add the broth, Marsala, heavy cream, thyme, ¼ teaspoon salt, and ⅛ teaspoon pepper; scrape any brown bits from the pan into the liquid. Bring to a boil, then reduce heat to medium and gently boil uncovered until the sauce reduces by about half, 10 to 15 minutes.

Add the chicken back to the pan along with any juices. Reduce the heat to low and simmer until the chicken is warmed through and the sauce thickens more, 2 to 3 minutes.

Sprinkle with parsley if using, and serve.