Thoroughly wash and chop the leeks, focusing on the white and light green parts. Slice into thin rounds and set aside.

If making your own broth, add dark green leek tops to a pot with water, fresh thyme, crushed garlic, and salt. Simmer for 30 minutes and strain.

In a large pot, melt butter over medium heat, add leeks, salt, and pepper. Sauté for 8 to 10 minutes until leeks are translucent.

Add potatoes, minced garlic, and broth to the pot. Bring to a simmer and cook for about 20 minutes until potatoes are fork-tender.

Remove thyme sprigs and blend the soup until smooth using an immersion blender. Return to pot.

Stir in milk or cream over low heat, adjusting seasoning with salt, pepper, or lemon juice as needed.

Ladle soup into bowls and garnish with chopped chives. Serve with crusty bread or grilled cheese.