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Cover egg with water in a small saucepan and bring to a boil, then simmer 3 minutes. Drain and cool under cold running water.
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Drain jarred asparagus, reserving liquid; coarsely chop, leaving tips whole.
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Blend chopped asparagus (not tips), egg, oil, vinegar, parsley, mint, sparkling water, and 3/4 cup asparagus liquid in a blender until smooth. Season with salt.
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Chill soup, covered, until cold, about 20 minutes.
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Divide asparagus tips, ham, and fresh asparagus among 4 shallow bowls, then pour in soup. Drizzle with a little additional oil.