Prepare the Leeks
Cut off the dark green parts of the leeks, leaving only the white and light green sections. Slice them into rounds, then rinse thoroughly under cold water to remove any dirt or sand.
Sauté the Leeks and Garlic
In a large pot or Dutch oven, melt the butter over medium heat. Add the leeks and cook until soft and fragrant, about 5-7 minutes. Add the minced garlic and cook for an additional minute, being careful not to let it burn.
Add the Potatoes and Broth
Add the diced potatoes, broth, and water to the pot. Stir to combine, then bring the mixture to a boil. Reduce heat, cover, and simmer for 20-25 minutes, or until the potatoes are tender.
Blend the Soup
Using an immersion blender, blend the soup until smooth and creamy. Alternatively, you can transfer the soup to a regular blender in batches and blend until smooth, then return it to the pot.
Add the Cream and Season
Stir in the heavy cream, and season with salt and pepper to taste. Heat gently until warmed through, but do not let the soup boil after adding the cream.
Serve and Garnish
Ladle the soup into bowls and garnish with chopped fresh chives or parsley for added color and flavor. Serve hot with a side of crusty bread.