Place the flour, ¾ teaspoon salt, and ¼ teaspoon pepper in a ziplock bag. Add the pounded chicken, seal the bag, and shake to coat evenly. Set aside.
Heat olive oil and 2 tablespoons butter in a large skillet over medium-high heat. Once the butter melts and bubbles, place the chicken in the skillet. Cook for 5-6 minutes, turning once, until golden brown and barely cooked through. Transfer to a plate.
Reduce the heat to medium and melt the remaining tablespoon of butter in the pan. Add mushrooms and cook for 3-4 minutes, stirring frequently, until browned.
Stir in shallots and garlic with ¼ teaspoon of salt. Cook for 1-2 minutes until fragrant.
Pour in the chicken broth, Marsala wine, heavy cream, and thyme. Use a wooden spoon to scrape any browned bits off the skillet bottom. Bring the mixture to a boil, then lower the heat and simmer for 10-15 minutes until the sauce reduces by half.
Return the chicken to the skillet, along with any juices on the plate. Reduce the heat to low and simmer for 2-3 minutes, allowing the flavors to meld and the sauce to thicken slightly.
Sprinkle with fresh parsley if desired, and serve warm. Pair with sautéed greens or cauliflower rice for a lighter meal.