Prepare the Chicken: Place each chicken breast on a cutting board, cover with plastic wrap, and slice horizontally to create four thin cutlets. Flatten them to about ½ inch thickness using a meat mallet or rolling pin.
Coat the Chicken: Mix the flour, ½ teaspoon of salt, and ½ teaspoon of pepper in a shallow dish. Dredge each chicken cutlet in the flour mixture, ensuring even coverage. Shake off the excess.
Cook the Chicken: Heat 1 tablespoon of butter and 1 tablespoon of olive oil in a large nonreactive skillet over medium heat. Add the chicken and cook for 3–4 minutes per side, or until golden brown and fully cooked (internal temperature of 165°F). Remove and cover loosely with foil. Repeat for all cutlets, adding oil if needed.
Sauté the Vegetables: In the same skillet, heat the remaining 2 tablespoons of butter. Add the mushrooms, onion, and remaining olive oil. Cook for 6–8 minutes, stirring frequently, until softened and lightly browned.
Build the Sauce: Stir in the garlic, thyme, and the remaining salt and pepper. Cook for 30 seconds until fragrant. Pour in the wine and bring to a boil. Scrape up any browned bits from the pan, then simmer for 7–10 minutes until the liquid reduces and thickens.
Finish the Sauce: Add chicken broth, stirring well, and continue simmering for another 10 minutes until the sauce thickens slightly. Stir in the heavy cream, reduce heat to medium-low, and simmer for 1–2 minutes more.
Combine and Serve: Return the chicken to the skillet, coating it in the sauce. Adjust seasoning with more salt or pepper as needed. Garnish with parsley before serving, if desired.