Season the cavity of the chicken with salt, pepper, and butter.
Add tarragon sprigs or dried tarragon to the cavity.
Truss the chicken and dry it thoroughly.
Rub the skin with another tablespoon of butter.
Heat oil and butter in a Dutch oven over medium heat.
Brown the chicken breast down for 2 to 3 minutes, then turn to brown all sides for about 10 to 15 minutes.
Remove the chicken and set aside.
Add onions and carrots to the Dutch oven and cook until softened, about five minutes.
Season the vegetables with salt and tarragon.
Salt the outside of the chicken and place it back in the Dutch oven on top of the vegetables.
Baste with the butter-oil mixture, cover with foil and lid.
Cook in the oven at 325°F until the internal temperature reaches 165°F, about one hour and 10 minutes.
