Trim the stems and dark green leaves off the leeks, leaving the white and light green parts. Slice them in half lengthwise and then chop across into small pieces.
Place the chopped leeks in a colander and rinse under cold water to remove any dirt or debris.
Heat the olive oil in a large pot or Dutch oven over medium heat. Add the leeks and sauté for 8 to 10 minutes, stirring occasionally, until softened but not browned. Add the minced garlic and cook for another minute.
Add the diced potatoes, salt, bay leaf, thyme, and vegetable broth to the pot. Increase the heat to high and bring to a boil. Reduce the heat to low, cover the pot, and simmer for 15 to 20 minutes, or until the potatoes are tender when pierced with a fork.
Remove and discard the bay leaf and thyme sprigs. Use an immersion blender to blend the soup to your desired creaminess. For a smoother texture, blend the soup in batches using a high-powered blender.
Serve the soup hot, garnished with freshly chopped chives and a sprinkle of black pepper.