In a shallow dish, stir together the flour and 1/4 teaspoon of the pepper. Dredge the chicken pieces in the seasoned flour.
In a large, heavy saucepan or Dutch oven, heat the oil over medium-high heat. Add the chicken and cook, turning once, until browned on both sides, about 5 minutes total. Transfer to a platter.
Add the shallot to the pan and saute until softened, about 1 minute. Add the mushrooms and saute until lightly browned, 3 to 4 minutes. Stir in the onions and saute until they begin to pick up some brown color, 2 to 3 minutes.
Stir in the stock and wine and deglaze the pan, stirring with a wooden spoon to scrape up any browned bits. Return the chicken pieces to the pan, and bring to a boil. Cover, reduce the heat to low, and simmer, stirring occasionally, until the chicken and vegetables are tender, 45 to 50 minutes. Stir vinegar, chopped thyme, teaspoon salt and the remaining 1/4 teaspoon pepper.
To serve, divide the vegetables among warmed shallow individual bowls. Top each portion with 2 pieces of chicken, 1 light meat and 1 dark. Garnish with thyme sprigs. Serve immediately.