01 - In a large pot, melt butter over medium heat. Add sliced leeks and chopped onion, sautéing for 5 to 7 minutes until softened but not browned.
02 - Incorporate minced garlic and cook for 1 minute more, stirring continuously to prevent burning.
03 - Add diced potatoes and vegetable broth to the pot. Bring the mixture to a boil, then reduce heat and simmer uncovered for 20 to 25 minutes, or until the potatoes are very tender.
04 - Remove the pot from the heat. Using an immersion blender, carefully puree the soup directly in the pot until it reaches a smooth consistency. Alternatively, blend in batches using a countertop blender.
05 - Stir in the milk or cream. Season the soup with salt and freshly ground black pepper to taste. Gently warm the soup over low heat if necessary, ensuring it does not boil.
06 - Ladle the creamy potato leek soup into serving bowls. Garnish each portion with finely chopped fresh chives before presenting.