Ina large pot, heat the butter or olive oil over medium heat. Add the chopped onion and minced garlic. Cook until the onion turns translucent and releases its fragrant aroma.
Add the sliced leeks to the pot. Cook for about 5-7 minutes, stirring occasionally, until the leeks are softened.
Add the diced potatoes to the pot. Pour in the vegetable or chicken broth, ensuring the potatoes are covered. Bring the mixture to a simmer.
Let the soup simmer for about 15-20 minutes, or until the potatoes are tender and can be easily pierced with a fork.
Using an immersion blender or regular blender (in batches), carefully blend the soup until smooth and creamy.
Return the blended soup to the pot over low heat. Stir in the milk to achieve the desired consistency. If the soup is too thick, you can add more milk or broth.
Adjust the soup's flavor to your liking by adding salt and pepper.. Keep in mind that the saltiness of the soup will depend on the type of broth used.
Continue to heat the soup gently for a few more minutes, allowing the flavors to meld.
Once heated through, ladle the Potato Leek Soup into bowls. Garnish with chopped chives or parsley for a burst of fresh flavor.