Slice the leeks lengthwise and rinse well. Dirt hides inside them. Cut into thin rounds. Let them drain. This step matters. Clean leeks make a clean tasting soup. Take your time here.

Place a large pot over medium heat. Add butter and olive oil. Let them melt together. This mix adds flavor. It also stops the butter from burning too fast.

Add chopped onion and leeks to the pot. Stir gently. Cook for about ten minutes. Do not rush. Let them soften slowly. They should look silky, not brown.

Add diced potatoes to the pot. Sprinkle in salt and black pepper. Add the bay leaf. Stir everything together. The potatoes will soak up the flavor at this stage.

Pour vegetable stock into the pot. Make sure everything is covered. Bring it to a gentle boil. Then lower the heat. Let it simmer quietly.

Let the soup simmer for twenty minutes. The potatoes should break easily with a spoon. The leeks will melt into the broth. Your kitchen will smell amazing now.

Remove the bay leaf. Use an immersion blender to blend until smooth. If using a regular blender, blend in batches. Be careful. Hot soup needs patience.

Return the soup to low heat. Stir in milk and cream. This gives the soup its soft texture. Taste and adjust seasoning. Add more salt if needed.

Ladle the soup into bowls. Garnish with chopped chives. Serve warm. Enjoy every spoon.