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Heat oven to 220C/200C fan/gas 7. For the tomatoes heat the oil in an ovenproof frying pan (if you don't have an ovenproof pan, use a standard frying pan then transfer to a roasting tray before putting in the oven) and fry cut-side down. Add the thyme and garlic and cook for 3 mins until coloured. Turn over the tomatoes, drizzle with vinegar, then roast in the oven for 20 mins until soft and caramelised. Leave to rest.
While the tomatoes are roasting, cook the lamb. Season the meat generously and heat the oil and butter in another ovenproof frying pan. Place the lamb, fat-side down, in the pan and scatter round the garlic and thyme. Brown the lamb really well, then turn over.
Baste the lamb with the pan juices, then transfer to the oven for 10 mins for lamb that is pink. If you prefer it more well done, give it 5 mins more. Leave the lamb somewhere warm to rest.
While the lamb is cooking, prepare the bean salad. Bring a large pan of salted water to the boil and cook the broad beans for a few mins. Scoop them out with a slotted spoon straight into a bowl of ice water, then slip them from their skins into a bowl. Boil the fine beans for 2 mins, then add the mangetout and cook both for 1 min longer. Drain, plunge into iced water (this keeps their bright green colour), leave for 1 min to cool down, then drain again.
Heat the oil in a frying pan, quickly cook the shallots for 1 min until starting to soften and colour, then add the fine beans and mangetout. Cook for about 1 min to heat through, then toss in the almonds and finally toss through the broad beans.
Scatter the basil over the tomatoes, saving a few of the very small leaves for garnishing. Make the dressing by simply whisking the oil and vinegar together. Carve the lamb into cutlets on a board with a very sharp knife (see tip, below). You are now ready to plate up.
Place the salad to one side of each plate. Line up three tomato halves down the other side of the plate. Lay three chops on top of the salad, slightly overlapping, with the bones pointing in the same direction. Drizzle the dressing around the plate and scatter the small basil leaves over the tomatoes.