01 - In a large soup pot, melt the unsalted butter over medium heat. Add the sliced leeks and diced onion, sautéing until softened but not browned, approximately 6–8 minutes.
02 - Stir in the minced garlic and cook for 1 minute until fragrant.
03 - Add the diced potatoes and vegetable broth to the pot. Bring the mixture to a boil, then reduce the heat to a simmer. Cook uncovered for 20 minutes, or until the potatoes are tender when pierced.
04 - Using an immersion blender, carefully purée the soup to your desired consistency, from chunky to completely smooth. Alternatively, transfer the soup in batches to a standard blender, process until smooth, and then return it to the pot.
05 - Stir in the whole milk or heavy cream. Warm the soup gently over low heat, ensuring it does not boil. Season generously with salt and freshly ground black pepper to taste.
06 - Ladle the hot soup into individual bowls. Garnish generously with finely chopped fresh chives and, if desired, a drizzle of extra virgin olive oil.