Heat 4 tablespoons oil in a large pot over medium-low heat. Add onions and cook, stirring occasionally, until translucent, 8–10 minutes. Add 1/2 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.
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Set a fine-mesh sieve over a large bowl; set aside. Working in batches, puree soup in a blender until very smooth. Strain through prepared sieve. (I put it all in a blender until smooth).
Stir remaining 2 tablespoons 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.
Divide asparagus tips and thinly sliced stalks among bowls; pour chilled soup over. Drizzle each with a few drops of oil.
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