In a large pot or Dutch oven, heat the butter and olive oil over medium heat. Once melted, add the yellow onion and sauté until it becomes translucent, about 5-7 minutes. Stir in the minced garlic and cook for an additional minute, until fragrant.
Add the trimmed and chopped asparagus to the pot. Sprinkle in the coarse salt and black cracked pepper. Sauté for about 5-8 minutes until the asparagus is bright green and slightly tender.
Pour in the low-sodium chicken broth and bring the mixture to a boil. Once boiling, reduce the heat and let it simmer for about 10-15 minutes, or until the asparagus is tender enough to blend.
Using an immersion blender, puree the soup until it reaches your desired consistency. For a smoother texture, blend it longer. If using a regular blender, allow the soup to cool slightly before blending in batches.
Return the blended soup to the pot over low heat. Stir in the heavy cream and shredded Parmigiano-Reggiano. Mix well until the cheese is melted and the soup is heated through. Taste and adjust seasoning if needed.
Ladle the Cream of Asparagus Soup into bowls. Drizzle with additional heavy cream for garnish and sprinkle with fresh chives or green onions to add a pop of color and flavor.