Heat the butter and the olive oil in a large pan, then add the parsnips and some seasoning. Put the lid on and sweat the parsnips for 8-10 minutes over a medium heat, stirring occasionally. Pour in the stock and bring to the boil. Reduce the heat and simmer for 15-20 minutes.
Blend the cooked parsnips and stock in a food processor or liquidiser until smooth. You may need to do this in two batches. For a really smooth soup, pass it through a fine sieve using the back of a spoon or ladle.
Pour the soup into a clean pan, stir in the cream and season to taste. Return to the boil, then turn down the heat and simmer until it is a fairly thick consistency.
As the soup is simmering, toast the chestnuts in a pan with a little oil and seasoning until lightly golden. Drain on kitchen paper.
Serve the soup in warmed bowls, float a little pesto in the middle, then scatter over the toasted chestnuts.
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