Prepare the leeks and potatoes. Trim the roots and dark green tops from the leeks, keeping the white and light green parts. Slice them thinly and rinse thoroughly under running water, separating the layers to remove any trapped dirt. Peel and dice the potatoes into evenly sized pieces.
Sauté the leeks. Heat the butter and olive oil in a large pot over medium-low heat. Add the sliced leeks and cook for 7 to 8 minutes, stirring occasionally, until softened. Keep the heat moderate so the leeks become tender without browning.
Add the garlic and herbs. Stir in the minced garlic and thyme. Cook for about 30 seconds, just until fragrant. Avoid browning the garlic, as it can develop a bitter taste.
Simmer the potatoes. Add the diced potatoes, vegetable broth, bay leaf, salt, and black pepper. Stir to combine and bring the mixture to a gentle boil. Reduce the heat, cover partially, and simmer for 15 to 20 minutes, or until the potatoes are completely tender when pierced with a fork.
Blend until smooth. Remove the bay leaf. Use an immersion blender to blend the soup directly in the pot until smooth and creamy. Alternatively, carefully transfer the soup to a countertop blender in batches, leaving room for steam to escape and following the blender manufacturer's instructions for hot liquids.
Add the cream. Stir in the heavy cream and warm the soup gently over low heat for 2 to 3 minutes. Avoid boiling it after adding the cream. Add the lemon juice and taste, adjusting the salt and pepper as needed.
Garnish and serve. Ladle the soup into warm bowls and sprinkle with chopped fresh chives. Add a little freshly ground black pepper or a small drizzle of cream, if desired. Serve immediately with crusty bread or your favorite side dish.