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Trim all the veg and peel the carrots. Boil a large pan of water and have a bowl of heavily iced water ready. Working in batches, cook the turnips for 3 mins, scoop into the iced water, then scoop out to drain. Repeat the process, cooking the carrots for 4 mins, the leeks for 5 mins, the peas and broad beans together for 1 min and finally the onions for 8-10 mins. Use a clean cloth to rub the skins off the turnips. Put all the vegetables in separate piles on a plate. TIP: Use the timings for cooking the baby vegetables only as a guideline, as they can vary in size. To be sure the vegetables are cooked properly, add a few more than the required amount to the water for you to test as they cook.
Slice the lamb fillets into finger-thick pieces, then season generously with salt and freshly ground pepper.
Heat 2 tbsp of oil in a large non-stick frying pan, then fry the lamb pieces for 2 mins on each side for rare or 3 mins on each side for medium. Tip the lamb into a colander with a bowl underneath to catch the juices, then leave in a warm place. TIP: When you cook lots of pieces of meat together, place them in the pan like points on a clock face – this makes it easy to remember which needs turning and removing from the pan first.
Place the pan back on the heat and tip in the wine. Boil vigorously until reduced to a sticky syrup, then pour in 200ml of the chicken stock and any lamb juices from the bowl. Boil down until reduced by about half, then whisk in the butter. Season with salt and pepper to taste, then stir in the chopped tarragon. Pour the sauce into a small bowl, then wipe out the pan with kitchen paper.
Heat a drizzle of oil and add the turnips and onions. Sizzle until starting to brown, then sprinkle over the sugar and a pinch of salt. Cook, shaking the pan constantly, until the veg are caramelised. Add carrots, leeks and balsamic, bubble for a moment, then add the stock. Bring to the boil, add the peas and broad beans, then boil for a few mins until all the liquid has nearly evaporated. Turn off heat.
To serve, dress each bowl by placing a few pieces of lamb on the base, spooning the smaller vegetables around the lamb and balancing the carrots and leeks on top. Pour the hot sauce over everything, scatter with tarragon leaves and finally drizzle with olive oil.