Prepare the chestnuts first in the case of starting from the raw fruit, roasting them in the oven, over the fire, in the microwave or by cooking them. Peel well, also removing the inner layer. Chop the washed leek and spring onion.
Heat the butter in a saucepan and poach both vegetables with a pinch of salt, letting them cook slowly. When they are transparent add the chestnuts with the peeled and chopped potato, season lightly and give a few turns. Add the vinegar, sauté for a couple of minutes and add the thyme.
Cover with the broth or water, bring to a boil, cover and simmer for about 30 minutes, stirring occasionally. When the vegetables are tender, shred very well. Add a splash of liquid cream to taste and correct salt.
Serve with a poached egg with the honeyed yolk, sweet paprika, fresh parsley and a splash of good extra virgin olive oil.