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Heat the oil in a large pan and sweat the onion, carrots, celery, fennel, turnip, celeriac and parsnip for 5-10 mins. Pour over 1.6 litres of boiling water, season and simmer for 10-15 mins, until the vegetables are tender. At the last moment add the peas and the beans and cook for a further 1 min. Taste and season with salt and pepper as necessary.
Meanwhile, make the pesto sauce. Blanch the basil leaves in boiling water for 5 secs, then remove and cool under cold running water. Pat dry and purée all the ingredients in a liquidiser. Taste and season, then reserve. Keep in the fridge until required. When ready to serve, pour the soup into a large, warmed tureen, top with some of the pesto sauce and serve to your guests. Place Parmesan and croûtons on the table so guests may help themselves.