In a medium enameled cast-iron casserole, melt 1 tablespoon of the butter. Add the shallot, carrot and celery and cook over moderate heat, stirring occasionally, until the vegetables are softened and beginning to brown, about 10 minutes. Season with salt and pepper.
Add the remaining 1 tablespoon of butter to the casserole and let melt. Increase the heat to moderately high and sprinkle 2 tablespoons of the flour over the mixture. Cook, stirring, until the flour begins to brown, about 3 minutes. Stir in the wine and bring to a boil. Add the chicken broth about 1/2 cup at a time, allowing the sauce to return to a boil and thicken between additions. Once all the broth has been added, stir in the chicken and season with salt and pepper. Keep the stew warm over moderately low heat.
In a medium bowl, mix the remaining 1 cup of flour with the baking powder, chives and 1/2 teaspoon of salt. Gently stir in the buttermilk and egg until the mixture is evenly moistened. Drop generous tablespoons of batter over the chicken stew. Cover the casserole and cook over moderately low heat until the dumplings are puffed and cooked through, about 12 minutes. Garnish with the parsley and serve immediately.