Clean the leeks: Rinse them under cold running water, ensuring all dirt is removed. Slice them thinly.

Melt the butter in a large pot over medium heat.

Add the sliced leeks and chopped onion to the pot. Sauté until softened, about 5-7 minutes. Stir occasionally to prevent browning.

Once the leeks and onion are tender, add the diced potatoes and stir together for another minute.

Add the vegetable or chicken broth to the pot, ensuring the potatoes and leeks are submerged.

Stir in the salt, black pepper, and thyme. Bring the mixture to a boil.

Reduce heat and let the soup simmer for about 20-25 minutes or until the potatoes are tender.

Using an immersion blender, blend the soup until smooth. For a chunkier texture, blend until your desired consistency is reached.

Stir in the whole milk or cream to make the soup rich and creamy. Heat gently but do not boil.

Taste the soup and adjust seasonings as necessary. Remove from heat when ready.

Serve hot and garnish with fresh chives if desired.