Celery and parsley roots peel, wash and roughly dice. Peel and finely dice onions and garlic. Wash rosemary, dab dry, pluck needles and chop.
Heat the oil in a large pot. Brown the onions and garlic in it. Stir-fry the celery and parsley roots for 6-8 minutes. They should not take on any colour. Steam rosemary for the last 2 minutes.
Deglaze with wine and 1 1⁄2 l water, stir in broth and bring to the boil. Add chestnuts and simmer covered for about 15 minutes.
Coarsely chop the walnut kernels. Heat a pan without fat. Roast the nuts, 1 tablespoon of water and 1 teaspoon of sugar while stirring and lightly caramelise. Immediately put it on a piece of baking paper and let it cool down.
Sort the lamb's lettuce, wash and drain. Pluck the leaves.
When the vegetables are soft, pour on the cream and finely puree the soup with a hand blender. Season to taste with salt and pepper. Serve with lamb's lettuce leaves and roasted walnut kernels.