Put your onion, garlic, carrot and basil stalks into a large pot with a couple of lugs of olive oil. Cover the pan and simmer gently without colouring for 20 minutes, stirring every couple of minutes.
Whisk together the cream, vinegar and egg yolks in a small bowl and put to one side.
While the vegetables are simmering, drop the tomatoes into boiling water for 30 seconds, then remove the skins and roughly chop the flesh. Add these to the veg, then pour in the stock and simmer for a further 20 minutes with the lid on.
Leave to cool a little then puree the soup in a liquidiser.
Put back into pan, bring back to a simmer and season carefully with salt & pepper.
Just before serving whisk in the cream mixture, but do not let it return to the boil. Serve straight away sprinkled with a few torn up basil leaves.