Optional: To prepare the leek top broth, slice off the dark green tops from the leeks, rinse thoroughly, and combine in a large stockpot with water, thyme sprigs, smashed garlic, salt, and peppercorns. Bring to a vigorous boil over high heat, then lower the flame and let it simmer for 30 minutes. Strain, reserving 6 cups for the soup. Extra broth may be refrigerated or frozen for future use.

To make the soup, finely chop the white and pale green parts of the leeks and wash them thoroughly to remove any grit. In a large Dutch oven or heavy-bottomed pot set over medium heat, melt the butter. Add the leeks, sprinkle with salt, and season generously with freshly ground black pepper. Cook, stirring from time to time, for 8 to 10 minutes until the leeks turn tender; lower the heat if they begin to brown too quickly.

Add the chopped potatoes, grated garlic, broth, and bundled thyme. Bring to a gentle simmer and cook for 20 minutes until the potatoes become very soft. Remove the thyme bundle. Using an immersion blender or carefully transferring to a countertop blender, purée the soup until silky smooth. Return the puréed soup to the pot, pour in the milk or cream, and stir until incorporated. Taste and adjust seasoning, adding lemon juice if desired.

Serve hot, garnished generously with chopped fresh chives.