Preheat the oven to 180C/350F/Gas 4.
<strong>Make the turnip gratin:</strong> bring two saucepans of salted water to the boil.
Add the sliced turnip to one pan and the sliced potatoes to the other. Bring both pans to the boil and cook briskly for 2 minutes; drain the turnip and potato separately and pat dry.
Whisk together the double cream, soured cream and Dijon mustard. Season with salt and pepper. Mix in the spring onions.
Toss the blanched turnip and potato slices with the cream mixture, keeping the turnip and potato separate.
In a greased baking dish or roasting tray, layer alternate layers of turnip and potato.
Sprinkle over the gruyère cheese and bake for 35 minutes.
<strong>Prepare the duck:</strong> place the honey, vinegar and the water in a saucepan. Bring to the boil, then reduce to a simmer.
Add each duck breast to the simmering honey mixture, allowing 20 seconds for each side.
Remove the duck breasts, season with salt and pepper, place on a clean tea-towel and chill in the refrigerator for 15 minutes.
Heat a lightly oiled griddle pan until hot. Place the duck breasts on the griddle skin-side down and sear briefly.
Transfer the duck breasts to a rack in a roasting tray and roast in the oven, alongside the gratin, for 10 minutes.
Remove from the oven and rest for 10 minutes. Just before serving, return the duck to the hot oven for 2-3 minutes to reheat.
While the duck is resting, prepare the beetroot jus. Heat the red wine jus in a saucepan and add in the diced beetroot. Cook briefly until the beetroot is heated through, season with salt and pepper, remove from direct heat and keep warm.
Meanwhile, bring a large saucepan of salted water to the boil. Add in the bok choi and cook for 5 minutes; drain and pat dry.
<strong>To serve:</strong> cut four squares of turnip gratin and place a square on four warm serving dishes.
Place the bok choi on top of the gratin.
Slice each duck breast into 4-5cm1½-2in thick slices and arrange the slices over the bok choi.
Spoon the beetroot jus around the gratin. Garnish with fresh chervil and serve at once.
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