In a large pot, sauté leeks, onions, and celery in oil over medium heat for 5 minutes.
Meanwhile, peel potatoes and dice into 1/2” to 1” dices. Turn heat on high and add potatoes to pan along with 2 cups of the water (or vegetable stock or broth) and the seasonings.
Cover and bring to boil. When soup comes to a boil, reduce heat to medium and cook until potatoes are just tender, but not too soft (about 4 to 6 minutes, depending on the type of potato you use).
Meanwhile, remove and discard the tough ends from the asparagus. Cut spears into 1/2” to 3/4” pieces.
When potatoes are tender, remove 2 cups of the soup mixture from the pan and pour it into a blender. Set aside.
Stir remaining cup of water or vegetable broth and asparagus pieces into soup. Cook uncovered at a low boil for 5 minutes or until asparagus is tender-crisp.
When asparagus is tender-crisp, stir in blender mixture. Cook until thoroughly heated.
Notes
The dark green part of a leek can be very tough; therefore, you will want to only use the white, yellow, and very light green part of the leek (minus the root).