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