Start off with the brine. Bring 1.6 litres of water to the boil in a big saucepan and add the sugar and salt. Stir until the sugar and salt are completely dissolved, then take off the heat and leave to cool. Remove and discard the wing tips and giblets from the chicken and place it in the brine. Leave for two and half hours in the fridge. Once the brine has done its work, drain the chicken and pat it dry with kitchen paper. Place in a heavy-bottomed roasting dish.
Preheat the oven to 220°C (200°C fan). Melt 50g of the butter in a pan, then pour it over the chicken. Squeeze over the lemon, tuck the garlic cloves into the dish, then pour in the wine and stock. Pop the chicken in the oven and roast for 45 to 60 minutes, or until the juices run clear when you poke a skewer into the thigh. Keep an eye on the sauce in the tray and don’t let it over-reduce -you can always add a splash of water.
Once the chicken is in the oven, start on the potatoes. Cut them into 2cm cubes, then parboil in salted water for five minutes. Drain and let them dry off a bit, then ruffle by gently tossing in the colander. Melt the duck or goose fat in a roasting tray, then add the potatoes and shake them around so that they are covered in the fat. Roast for around 20 minutes, turning regularly, until golden all over. Add the garlic and rosemary or thyme to the roasted potatoes when they have reached the desired colour. Return to the oven for two minutes (keep an eye on the garlic as it may burn).
When the chicken is cooked, leave it to rest for ten minutes or so, loosely covered in foil. Transfer the stock and juices to a saucepan and bring to a simmer. Whisk in the remaining 50g of butter and the fresh tarragon. Serve the chicken with potatoes on the side and plenty of jus.