Wash leeks very carefully to remove all grit. I usually cut them horizontally and separate the rings to make sure no dirt remains. Thinly slice the white and light green parts when washed.
In a large soup pot over medium heat, melt the butter, add the prepared leeks, salt, and pepper. Cook, stirring regularly, until leeks are soft, wilted, and golden, about 8-10 minutes. Reduce heat if leeks are browning. Add garlic and cook, stirring for 1 minute more. While leeks are cooking, chop the potatoes.
Add potatoes, vinegar or lemon juice, chicken stock, fresh thyme, and bay leaves to the pot. Bring to a boil (uncovered) and once boiling, cover pot with a lid and turn heat to low. Simmer for 15–18 minutes or until potatoes are easily pierced with a fork. Turn off heat and let slightly cool, about 10 minutes. Pull out thyme and bay leaves; discard.
Use an immersion blender to blend or transfer in batches to a high-powered blender (don’t fill the blender above 1/2 way full or it could explode). Blend until very smooth (about 2 minutes for each batch), then return to pot. Pour in heavy cream and stir to warm through over low heat.
Taste and adjust seasonings, as needed. If it seems too thin, simmer until it thickens. If it's too thick, add a splash more stock. Ladle into bowls, garnish with chives and enjoy with croutons and/or some warm, crusty bread for dipping!