Preheat the oven to 220C/425F/Gas 7.
Mash the butter and two-thirds of the tarragon together in a small bowl and season with salt and freshly ground black pepper. Stir in the crushed garlic.
Place the chicken into a roasting tin and rub the herb and garlic butter all over the outside and inside of the chicken.
Pour the wine into the roasting tin and place the tin into the preheated oven. Roast for 15 minutes, then turn the heat down to 190C/375F/Gas 5 and roast for a further 40 minutes, basting the bird from time to time with the juices from the pan, adding a little more wine or water if the tin becomes too dry.
When the chicken is golden-brown and cooked through (pierce the chicken at the thickest part of the thigh - the juices should run clear), turn the oven off and leave the chicken to rest for ten minutes before carving.
Add the remaining chopped tarragon to the roasting juices in the tin, stir well and season to taste with salt and freshly ground black pepper.
To serve, carve the chicken and pile the slices onto serving plates. Drizzle over pan juices and serve immediately.