Wash and slice the leeks into small slices.
Place a heavy bottom soup pot on medium low flame. Add the olive oil and the sliced leeks with a pinch of sea salt. Sautee the leeks slowly for about 15 minutes until they start to get some color. Reserve some for garnish at this point if desired.
Stir in the potatoes, carrot, thyme and bay leaf. Cover with the stock or water and bring to a simmer.
Cook on gentle flame until the potatoes and carrots are tender, about 15 minutes.
Discard the bay leaf and thyme sprigs and using a hand held blender purée the soup to your liking.
Meanwhile add a drop of oil to a hot cast iron skillet and sear the andouille sausage until golden on both sides.
Ladle the soup into bowls and drizzle with some of the heavy cream. Top with the seared andouille sausage and and garnish with the reserved leeks, chives and thyme. Serve with freshly cracked black pepper on top and a drizzle of really good extra virgin olive oil.