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 40 minutes total. Transfer the chicken to a platter and loosely tent with foil while finishing 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.