Melt the duck fat in a saucepan, add the garlic, thyme and potatoes and season with sea salt and pepper. Transfer to a roasting tin and cook for 40–60 minutes, until they are golden brown and cooked through.
For the last 25 minutes of cooking, add the beetroot and cook, until soft. Allow to cool before peeling the skins and chopping into pieces.
To make the carrot purée, put the carrots in a saucepan and cover with water. Add a pinch of salt and bring to the boil. Reduce the heat and simmer for 15–20 minutes, until they are soft.
Drain the carrots, reserving a little of the cooking water. Blend them to smooth purée in a food processor, adding a splash of the cooking water. Pass the purée through a sieve into a bowl. Cover and set aside.
Cook the turnips in a pan of salted water for 10 minutes, until soft. Drain and set aside.
To cook the duck breasts, lower the oven temperature to 190C/170C Fan/Gas 5.
Score the skin of the duck breasts and season with salt and pepper. Put the duck breasts skin side down in a cold ovenproof pan over a very low heat. Cook for 8 minutes, until the fat has rendered and the skins are golden brown and crisp.
Flip over the breasts and cook for a further 2 minutes, until golden brown. Put the pan in the oven and roast for 3–4 minutes, until the breasts are cooked medium rare. They should be slightly springy when pressed and still pink in the middle. Leave to rest for 4–5 minutes, then cut into slices.
Heat the duck jus in a saucepan, until hot. Heat olive oil in a pan and fry the turnips and beetroot for a couple of minutes.
To serve, spoon the carrot purée onto four plates and top with the duck. Arrange the roast potatoes, beetroot and turnips around the duck and spoon over the jus.