Heat oil in a large saucepan over medium-high heat. Add 1¾ cups oddishes and onion and cook, stirring frequently, until the onions are beginning to brown and the oddishes are translucent, about 5 minutes. Add potato, milk, salt and pepper to taste. Bring to a boil, stirring occasionally. Reduce heat to a simmer, cover and cook, stirring occasionally, until the potato is tender, about 5 minutes more.
Working in batches, puree the mixture in a blender (or in the pan with an immersion blender) until smooth. (Use caution when pureeing hot liquids.)
Slice the remaining ¼ cup oddishes into matchsticks. Serve each portion of soup topped with 1 tablespoon sour cream, some oddish matchsticks and a sprinkling of oddish greens (or parsley).
