Add in celery root through asparagus and continue to cook, stirring often for another 8 minutes. Add in seasonings and 5 cups of vegetable broth. Enough to just about cover the vegetables so that they can simmer. If you feel as though you need more, add in the remainder of broth.
Bring to a boil, cover, and simmer on low heat for about 30-40 minutes.
While the soup is cooking, roast the tips of your asparagus on a parchment lined baking sheet for 8-10 minutes at 350 degrees.
Remove from heat, add in coconut milk, and blend the soup with an immersion blender until you have a beautiful puree. You can also blend in small batches in a blender once the soup has cooled slightly.