Pat the chicken dry with paper towels. Also, preheat the oven to 350°F. Set the oven-proof skillet over medium heat on the stovetop. Add the olive oil and 1 tablespoon of the butter. When the butter melts, swirl the pan to evenly coat the bottom. Then sprinkle the salt and pepper (a generous pinch of each, or to taste) evenly over the hot fats.
Arrange the chicken pieces in the skillet, skin-side down. Cook, undisturbed, until the pieces are deeply browned -- about 4 minutes per side. Transfer the chicken to a plate and set aside.
Melt the remaining butter in the skillet. Add the onion, carrots, and mushrooms, and stir briefly to insure that all are coated with the fats in the pan. Cook, stirring occasionally, until the vegetables color slightly -- about 10 minutes. Stir in the garlic, and cook it for just 1 minute.
Sprinkle the flour over the vegetables, and then stir gently until a thick paste develops -- about 1 minute. Add the wine, and stir gently and constantly as the liquid thickens and bubbles -- about 1 minute. Add the herb bundle. Stir in the chicken broth or stock, and bring to a boil. Boil until the liquid reduces and thickens slightly -- 4-5 minutes.
Return the chicken to the skillet, and then transfer the skillet to the preheated oven. Cook, uncovered, until the chicken is done and the liquids in the pan are thick enough to coat a spoon -- about 45 minutes. Remembering the skillet's handle is red hot from the oven (use mitts!), transfer the skillet to the stovetop or a wire rack.
Add the cream, and stir gently to incorporate. Then garnish with the fresh or dried parsley. Serve hot on a bed of cooked rice, mashed potatoes, or egg noodles.