In a blender, combine the cucumbers, yogurt, basil, olive oil, lemon juice, dill, garlic, salt, honey, and several grinds of pepper and blend until smooth. Chill for at least 4 hours.
Ladle into bowls and drizzle with olive oil. Garnish with the reserved cucumber slices, freshly ground black pepper, and fresh dill, mint, and/or chives. Serve cold.
