Step 1 Trim the green tops from the radishes and rinse them well, discarding any that are discolored or wilted. Blanch in a 1-quart saucepan full of rapidly boiling salted water just until they are tender, about 30 seconds. Drain and transfer to a bowl full of ice water to stop the cooking. Squeeze dry and chop coarsely. Reserve 2 tablespoons to make the oil.
Step 2 Quarter all but 4 of the radishes. In a food processor, coarsely grind the quartered radishes, most of the blanched tops, garlic and green onions.
Step 3Transfer the mixture to a large bowl or pitcher, add the buttermilk and sour cream and stir well to combine. Season aggressively with salt and white pepper. Chill for a couple hours.
Step 4While the soup is chilling, blend the reserved 2 tablespoons blanched radish tops with the olive oil until smooth and strain it through cheesecloth into a bowl. Do not press or squeeze the solids or the results will be cloudy.
Step 5To serve, taste and adjust seasoning for salt and pepper. Thinly slice the remaining radishes. Divide the soup among 6 chilled bowls. Float the sliced radishes on top and drizzle with a little of the green oil.