For the chicken: In a Dutch oven or large saucepan, heat the oil over high heat. Season the chicken with salt and pepper, to taste, and dust with flour. Cook the chicken, turning occasionally, until brown on all sides, about 8 minutes. Remove the chicken and set aside.
For the sauce: Heat the same pan used for the chicken over medium-high heat. Add the onions and garlic and season with salt and pepper, to taste. Cook, stirring frequently, until softened, about 5 minutes. Increase the heat to high. Add the wine and scrape up the brown bits that cling to the bottom of the pan with a wooden spoon. Return the chicken pieces to the pan. Add 3 cups of chicken broth and 1/2 cup of tarragon and bring the mixture to a boil. Reduce the heat to a simmer, cover the pan and cook, turning the chicken pieces over every 10 minutes, until cooked through, about 30 minutes total for the breasts and wings and 40 minutes total for the legs and thighs. Transfer the chicken to a platter and loosely tent with foil while finishing the sauce.
To finish the sauce: In a small bowl, mix together the remaining 1/3 cup chicken broth and flour. Whisk the flour mixture into the sauce until smooth. Bring the sauce to a simmer and cook, uncovered, until thickened, about 8 to 10 minutes. Stir in the butter and season with salt and pepper, to taste. Serve the sauce over the chicken and garnish with the remaining 2 tablespoons of tarragon.