In a large pot, melt the butter or heat the olive oil over medium heat. Add the sliced leeks and minced garlic, and sauté until they become soft and fragrant, about 5 minutes.
Pour in the vegetable broth and bring the mixture to a boil. Reduce the heat to low, cover the pot, and let it simmer for about 15-20 minutes, or until the leeks are tender.
Using an immersion blender or regular blender, carefully puree the soup until smooth. If using a regular blender, you may need to blend the soup in batches, as hot liquids can expand during blending.
Return the blended soup to the pot and place it back on the stove over low heat. Stir in the milk and heavy cream (or full-fat coconut milk) until well combined.
Gradually add the shredded cheese to the soup, stirring continuously until the cheese melts and the soup becomes creamy and smooth. Be patient, as it may take a few minutes for the cheese to fully melt.
Season the soup with salt and pepper to taste. Remember that cheese can be salty, so start with a little salt and adjust as needed.
Once the soup is heated through and well-seasoned, it is ready to be served. Ladle it into bowls and garnish with chopped fresh chives or parsley if desired.Enjoy the delightful flavors of this comforting cheese leek soup, perfect for a cozy evening or a gathering with loved ones!