Combine 2½ cups ice water, the cashews, olive oil, chickpeas, aquafaba, garlic, vinegar, and salt in a high-powered blender, such as a Vitamix.
Puree until smooth. If the soup seems too thick, add more ice water, a few tablespoons at a time, until you have reached your desired consistency.
Taste and add more salt if needed.
Chill the soup until very cold, 1 to 2 hours (or as long as overnight).
To serve, divide among serving bowls, drizzle with olive oil, and garnish with cashews or roasted chickpeas, if desired, and chive blossoms and/or mint leaves.