Prepare the leeks by cutting the root ends and the dark green portions of the stalks. Discard the dark green parts*. Wash the white and light green parts under running water to remove all the dirt. Cut them in half from end to end, then slice each leek in half-moons. Add them into a bowl with water, swish to remove any extra dirt, and skim off the clean pieces from the water.
In a large soup pot over medium-low heat, water sauté the leeks, celery, and the onion for about 7 minutes. Add the thyme, a pinch of salt, and the garlic cloves. Cook for one more minute, stirring regularly.
Add the potatoes, asparagus, vegetable broth, bay leaves, salt and pepper. Stir to combine, bring to boil, and cover the pot, reducing the heat to medium-low. Simmer the soup for 20 to 25 minutes until the potatoes and asparagus are soft and tender.
Remove some veggies to keep some larger pieces for topping the soup when serving it. Place them in a bowl and set them aside.
Remove and discard the bay leaves.
Using an immersion blender, puree the soup until creamy and smooth**.
Taste the soup and adjust the seasoning with salt and pepper.
Serve warm topping with some reserved veggies, drizzle some lemon juice and add fresh parsley. You can also garnish with vegan sour cream.