Place the chicken in a small roasting pan. Slip your fingers between the skin and breast to make a pocket and insert the 4 slivers of butter and tarragon leaves. Pull the skin back in place. Rub both sides of the bird lightly with the softened butter. Season both sides and inside the cavity with salt and pepper. Arrange the onions and remaining herbs inside the cavity and tie the legs together with kitchen twine.
Roast for 20 minutes breast-side down. Turn the bird and roast for 20 minutes breast side-up. Lower the temperature to 375 degrees F, turn the bird breast-side down, and roast until golden brown, the juices run clear and an internal thermometer inserted into the thickest part of the thigh registers 170 degrees F., about 30 to 45 minutes more. Remove from the oven and transfer to a platter to rest for 15 minutes. Tent to keep warm.
Spoon off the fat from the pan juices, and place the roasting pan on 2 stovetop burners over medium heat. Add the chicken stock and bring to a boil, stirring to deglaze the pan, and scrape up any brown bits from the bottom of the pan. Simmer until reduced to about 3/4 cup, about 3 minutes.
Make a slurry using the white wine and cornstarch. Whisk the slurry into the hot sauce and bring to a boil to thicken. Remove from the heat and season, to taste, with salt and pepper. Transfer to a gravy boat or decorative bowl.
Carve the chicken and serve with the Herbed Rice Pilaf with Peas, and pan gravy.
Herbed Rice Pilaf with Peas:
Melt the butter in a medium saucepan over medium-high heat. Add the onions and cook, stirring, for 3 minutes. Add the garlic and cook, stirring, for 30 seconds. Add the rice, salt, and pepper, and cook, stirring, until the rice is glassy, about 2 minutes. Add the stock and bring to a boil. Reduce the heat to low, cover, and cook until almost all the liquid is absorbed, about 10 minutes. Add the peas without stirring. Cover and continue to cook until the liquid is absorbed, about 4 minutes. Remove from the heat and let sit covered without stirring for 15 minutes.
Fluff with a fork. Add the parsley and mint and stir to combine. Adjust seasoning to taste and serve.