01 - In a large soup pot, melt the butter over medium heat. Add the sliced leeks and diced onion, then sauté 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 thoroughly tender.
04 - Using an immersion blender, purée the soup to achieve your desired consistency; you may leave it slightly chunky or blend until completely smooth. Alternatively, transfer the soup in batches to a standard blender, purée, and then return it 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 finished soup into individual bowls. Garnish liberally with chopped fresh chives and a drizzle of extra virgin olive oil, if desired.