Prepare the leeks: Trim the root and dark green tops of the leeks. Slice the white and light green parts thinly. Thoroughly rinse the slices in cold water to remove any sand.
Sauté the leeks and garlic: In a large pot, melt the butter over medium heat. Add the leeks and sauté until softened, about 5-7 minutes. Add the garlic and cook for an additional minute.
Add potatoes and broth: Stir in the diced potatoes and vegetable broth. Bring to a boil, then reduce heat and let simmer for 20-25 minutes, or until the potatoes are tender.
Blend the soup: Use an immersion blender to purée the soup until smooth. Alternatively, you can transfer the soup to a blender in batches.
Stir in the cream: Once blended, return the soup to low heat and stir in the heavy cream. Season with salt and pepper to taste. Let the soup heat through for 3-5 minutes.
Finish and serve: Ladle the soup into bowls and garnish with fresh thyme or a swirl of extra cream. Serve immediately.
