Trim off the dark green tops and roots from the leeks, then slice the white and light green parts into thin rings. Rinse thoroughly in cold water to remove any grit, then drain well.
Peel the potatoes and cut them into 1-inch cubes, keeping them ready for cooking.
Heat the large pot over medium heat and add the butter or olive oil. Once melted and shimmering, add the sliced leeks.
Cook the leeks gently, stirring often, for about 8-10 minutes until they become translucent and fragrant, with a slight golden hue at the edges.
Add the minced garlic to the pot, stirring for about 1 minute until fragrant, but not browned.
Stir in the cubed potatoes, coating them with the leek mixture, then pour in the warm vegetable broth.
Increase the heat to high and bring the mixture to a gentle boil. Once bubbling, reduce the heat to low, cover the pot, and let simmer for about 20 minutes until the potatoes are very tender.
Use an immersion blender directly in the pot, or carefully transfer the soup in batches to a blender, blending until smooth and velvety. Return to low heat if needed.
Stir in the heavy cream and season with salt and pepper to taste. Warm through for another 2-3 minutes, stirring gently.
Taste and adjust seasoning as needed, then ladle the soup into bowls. Garnish with chopped herbs or a drizzle of cream if desired.
