Step 1

Heat 4 Tbsp. oil in a large pot over medium-low heat. Add onions and cook, stirring occasionally, until translucent, 8–10 minutes. Add ½-inch asparagus pieces and season with salt and pepper. Cook until asparagus is bright green and tender, 4–5 minutes. Add broth, increase heat to high, and bring to a boil. Reduce heat to medium and simmer until asparagus is tender, 8–10 minutes. Add spinach and cook, stirring occasionally, until wilted, about 2 minutes. Let mixture cool slightly.

Step 2

Set a fine-mesh sieve over a large bowl; set aside. Working in batches, purée soup in a blender until very smooth. Strain through prepared sieve. Stir remaining 2 Tbsp. oil into soup; season to taste with salt and pepper. Cover and chill until cold, at least 3 hours. DO AHEAD: Can be made 1 day ahead. Keep chilled.

Step 3

Divide asparagus tips and thinly sliced stalks among bowls; pour chilled soup over. Drizzle each with a few drops of oil.