Using tongs, remove and discard all but one quarter of the onion. Using an immersion blender, puree cauliflower mixture. (Or cool slightly. Transfer mixture in batches to a food processor or blender. Cover and process or blend until smooth, removing cap from lid and holding a folded kitchen towel over opening in lid. Return soup to pan.) Add half-and-half, the 1 teaspoon salt, and the white pepper. Heat through, stirring occasionally. Season to taste with additional salt.