01 - In a large pot or Dutch oven over medium heat, melt the unsalted butter. Add the sliced leeks and chopped onion, sautéing for 5 to 7 minutes until softened and translucent, ensuring they do not brown.
02 - Stir in the minced garlic and cook for an additional 1 minute, stirring frequently until fragrant.
03 - Add the diced potatoes and pour in the vegetable broth. Bring the mixture to a boil, then reduce the heat to low, cover partially, and simmer gently for 20 to 25 minutes, or until the potatoes are exceptionally tender.
04 - Remove the pot from the heat. Carefully use an immersion blender to purée the soup directly in the pot until it achieves a completely smooth, velvety consistency. Alternatively, transfer the soup in batches to a standard blender and process until smooth, then return to the pot.
05 - Stir in the whole milk or heavy cream. Return the pot to very low heat and warm gently, taking care not to bring the soup to a boil, which could cause the dairy to curdle. Season generously with salt and freshly ground black pepper to your preferred taste.
06 - Ladle the hot, creamy potato leek soup into individual serving bowls. Garnish each portion with a sprinkle of fresh, finely chopped chives before serving immediately.