Pour oil into a heavy bottomed soup pot. Place over medium heat.
When oil is hot and shimmering, add onion and celery. Sauté until they begin to sweat and soften, about 5 to 7 minutes.
Add chopped asparagus (reserve the tops), peas and garlic. Cook 1 minute more.
Add vegetable stock. Bring to a boil. Partially cover the pot, reduce heat to medium-low and simmer until the vegetables are just tender, about 10 minutes.
Remove soup from heat.
Use an immersion blender to purée ingredients to your preferred consistency (Note - At Au Bon Pain, they keep a little definition to the ingredients so that the soup isn’t complete smooth. If you don’t have an immersion blender you can puree the soup in batches in a blender. Use caution - hot ingredients expand when they’re blended (because of the steam).)
Return soup to low heat.
Add heavy cream, chives, Parmigiano-Reggiano and fresh lemon juice. Mix well.
Taste. Add salt and pepper, to taste.
Bring soup to a low simmer for five minutes.
Serve hot with reserved asparagus tops.