1. Preheat oven to 350° F. Trim any excess skin or fat from chicken. Cut the breastbone away from breast meat, and discard bone. Leaving skin intact, cut each breast into 2 pieces. Rub the chicken with salt and pepper. In a large Dutch oven, add 2 tablespoons of oil over high heat. Add the chicken. Cook on both sides until golden and light brown. Reduce heat to low. Cover the chicken, and cook for 5 minutes. Remove the lid, and turn the chicken. Cover, and cook for 5 more minutes. Uncover, and pour in the brandy. Turn heat to high. Cook until brandy is reduced and nearly evaporated. Pour in the wine, 1 cup of broth, tomato paste, garlic, bay leaves and 8 sprigs of thyme. Stir to completely combine the ingredients. Cover again, and place in oven for 30 minutes.
2. In a skillet, add remaining oil over medium heat. Add onions. Sauté for 2 to 3 minutes until golden and beginning to brown. Add remaining 1/2 cup of broth to pan. Turn heat to low, and cover skillet. Cook for 10 minutes. Remove onions from skillet, and set aside. Wipe out the skillet. Add 2 tablespoons of butter over medium heat. Add mushrooms. Sauté for 5 to 7 minutes until golden. Remove, and set aside.
3. Remove chicken from oven, and take out of the sauce. Discard thyme stems and bay leaves. Bring sauce to a simmer over low heat. In a small bowl, combine the remaining 2 tablespoons of butter and the flour. Mash together with a fork until thoroughly combined. Whisk the mixture into the sauce. Continue to whisk until butter is melted and flour is incorporated. Let simmer for 2 minutes to thicken.
4. Stir in the onions and mushrooms. Cook on low for 5 minutes. Add the chicken back to the sauce, and turn off the heat. Cover until ready to serve, or place in a 200° F oven to hold for longer than 15 minutes. When ready to serve, garnish with chopped parsley and additional thyme stems.

Per Serving: Calories: 680, Fat: 37 g (12 g Saturated Fat), Cholesterol: 125 mg, Sodium: 790 mg, Carbohydrates: 23 g, Fiber: 3 g, Protein: 30 g.