Heat the olive oil in a large pot over medium heat. Add the chopped onion, celery, and carrot. Sauté until the onion becomes translucent and the vegetables begin to soften, about 5 minutes.
Cut the asparagus into 1-inch pieces, reserving a few tips for garnish if desired. Add the minced garlic, diced potatoes, asparagus (excluding the tips) to the pot and cook for another 5 minutes, stirring frequently.
Pour in the vegetable broth, cover the pot, and bring to a boil. Reduce the heat and simmer for about 20 minutes, or until all the vegetables are tender.
Using an immersion blender or transferring the mixture to a blender, purée the soup until smooth and creamy. If using a blender, be sure to let the soup cool slightly before blending and work in batches if necessary.
Return the soup to the pot (if using a blender) and stir in the plant-based milk, tamari, and nutritional yeast. Heat the soup gently over low heat, stirring occasionally, until warmed through. Season with salt and pepper to taste.
While the soup is heating, you can lightly steam the reserved asparagus tips for garnish, if desired.
Once heated, ladle the vegan cream of asparagus soup into bowls. Garnish with steamed asparagus tips and fresh herbs of your choice.