Slice the leeks and rinse thoroughly to remove any dirt. Set aside.
- In a large pot, melt butter (or heat olive oil) over medium heat. Add sliced leeks and cook, stirring occasionally, for 8-10 minutes, until soft and translucent. Add minced garlic and cook for another 1-2 minutes.
- Add diced potatoes, vegetable broth, thyme, and black pepper to the pot. Bring to a boil, then reduce heat and simmer for 20-25 minutes, until the potatoes are tender.
- Blend the soup using an immersion blender until smooth and creamy. Alternatively, transfer the soup in batches to a blender (be cautious as it’s hot!). For a chunkier texture, blend only half the soup and leave some pieces intact.
- Stir in heavy cream (or half-and-half) to achieve a creamy consistency. Adjust seasoning with salt and pepper to taste.
- Ladle the soup into bowls, garnish with fresh parsley or chives, and serve with crusty bread for dipping.