Prepare the vegetables: Peel and dice the potatoes. Slice the leeks, using only the white and light green parts. Mince the garlic.

Sauté the leeks: In a large pot, melt the butter over medium heat. Add the leeks and garlic, sautéing until soft and fragrant, about 5 minutes.

Add the potatoes: Stir in the diced potatoes and fresh thyme. Cook for another 2 minutes to let the flavors meld.

Pour in the broth: Add the vegetable broth to the pot. Bring to a boil, then reduce heat and simmer until the potatoes are tender, about 20 minutes.

Blend the soup: Use an immersion blender to puree the soup until smooth. Alternatively, transfer in batches to a blender and blend until smooth.

Add the cream: Stir in the heavy cream and season with salt and pepper to taste. Heat through for another 2-3 minutes.

Garnish and serve: Ladle the soup into bowls and garnish with fresh chives. Serve hot with crusty bread.