Wipe and trim the chicken joints. Season well with salt and pepper. Wash the tarragon, strip the leaves from the stems and chop finely. Then, using a knife, cream the butter with the tarragon on the chopping board.
Place the chicken joints in a baking or roasting tin, skin side upwards. Spread the tarragon butter on each one. Place in the centre of a moderate oven (180°C., 350°F., Gas Mark 4) and cook for 1 hour. Baste the joints several times with the butter as it melts in the roasting tin. About 5 minutes before cooking time is completed, pour the cream over the chicken joints and return to the oven.
To serve, lift the chicken pieces out on to a hot serving plate. Stir the cream and butter in the roasting tin to blend the juices together. Check the seasoning, adding a little salt if necessary. Spoon over the chicken joints and serve.