Cut the crusts off the stale white bread and tear or cut into rough chunks. Soak in cold water for 10-20 minutes until soft and saturated. Do not drain.
Peel the cucumber and chop into chunks. Peel the garlic cloves, removing the green germ from each if present. Measure out the remaining ingredients.
Add almonds, garlic, soaked bread with liquid, cucumber, grapes, sherry vinegar, and salt to the blender. Blend on high for 60-90 seconds until smooth, scraping down sides as needed.
With the blender running on low, slowly drizzle in the olive oil in a thin, steady stream. This emulsifies the oil into the soup. Once incorporated, blend on high for another 20-30 seconds.
With the blender running on low, slowly add the additional cold water a little at a time until the soup reaches a pourable, creamy consistency. Taste and adjust seasoning with salt or more sherry vinegar.
(Optional) Set a fine mesh strainer over a large bowl and pour the soup through. Use a spatula to press the soup through, leaving behind any solids. This creates a silkier texture.
Transfer the soup to an airtight container or pitcher. Refrigerate for a minimum of 2 hours, or ideally overnight, to allow the flavors to meld and deepen.
Ladle the chilled soup into bowls. Garnish with a drizzle of olive oil, halved green grapes, toasted almond slivers, and fresh mint leaves if desired. Serve immediately while very cold.