To a large soup pot, heat the olive oil or butter over medium heat. Add the leeks and cook for 5 minutes, lowering the heat if necessary, to prevent burning. Then add the garlic and cook for 5 more minutes, stirring occasionally, until the leeks are lightly golden and the garlic releases its flavor.
Add the potatoes, white beans, bay leaves, thyme strings, salt, black pepper, and vegetable broth. Stir and bring to a boil. Simmer over medium-low heat for 15-20 minutes with a lid on, until the potatoes are very soft.
Take out the thyme and the bay leaves and use an immersion blender to blend the soup. If you are using a standard blender, make sure to let the soup cool a bit before handling. Fill the jar only halfway through and leave the hole in the lid open to allow steam to escape. Blend until creamy and pour into a clean pot.
Add this point, you can taste and adjust the flavor, and the thickens of the soup. If it's too thick, add more vegetable broth, one cup at a time, until you reach the desired consistency.
Serve with a drizzle of olive oil and some fresh thyme on top. If you're feeling extra fancy you can top it with some dairy-free cheese as well (optional). Enjoy!