Peel the cucumber and cut into rough 1-inch chunks. Halve the avocados, remove the pits, and scoop the flesh into the blender. If your cucumber is particularly seedy, scoop out and discard the seeds first for a smoother texture.

Add the cucumber, Greek yogurt, cold water, lime juice, garlic, salt, pepper, and olive oil to a blender. Blend on high until completely smooth and creamy, about 60 seconds, stopping to scrape down the sides once. The soup should coat the back of a spoon; if too thick, add cold water 1 tablespoon at a time.

For an ultra-silky texture, pour the soup through a fine-mesh strainer into a bowl, pressing gently with the back of a ladle. This step is optional but removes any remaining cucumber skin or fiber.

Cover and refrigerate for at least 30 minutes to allow flavors to meld and the soup to chill thoroughly. The soup can be made up to 4 hours ahead; press plastic wrap directly onto the surface to prevent browning.

Taste and adjust seasoning with more salt or lime juice as needed. Ladle into chilled bowls, drizzle with olive oil, and scatter fresh dill over top. Add microgreens or thin radish slices if using. Serve immediately.