Heat oil in a large saucepan over medium or medium-high heat. Add onions and radishes. Stir frequently and cook until onions and radishes are translucent (6-8 minutes).
Add potatoes, milk, salt, and pepper. Bring to a boil, reduce heat to a simmer, cover, and cook until the potatoes are tender. Stir occasionally to avoid burning.
Puree with an immersion blender until smooth. Taste and add salt and/or pepper if needed.
Top each serving with some radish matchsticks and a dollop of sour cream. Sprinkle some greens on top.