In a large (6-quart) pot, heat the butter on medium-high heat until it melts and starts to foams up. Continue to heat until the butter just begins to brown.
Immediately toss in the sliced leeks, carrots, and celery. Stir to coat with the butter. Cook until the leeks are translucent and slightly wilted.
Put the vegetable or chicken stock, potatoes, leeks, celery, bay leaf, and thyme in a large pot and sprinkle with salt and pepper. Boil until the potatoes are soft, 15 to 20 minutes.
Remove the bay leaf.
Blend about ½ the soup depending on how chunky you want the soup with an immersion blender (or in batches in a blender or food processor) and return to the pan.
Add the Half and Half or cream and simmer until the soup has thickened, about 20 minutes. Add more salt to taste. Serve garnished with chopped fresh chives.