Peel onion. Cut lengthwise into quarters, keeping the root end intact; set aside. In a Dutch oven or large saucepan cook bacon over medium heat until crisp. Remove bacon and drain on paper towels. Crumble bacon; set aside.
Drain bacon drippings from pan, reserving 1 tablespoon drippings in pan. Add onion and paprika to the reserved drippings. Cook over medium heat for 3 minutes, stirring paprika constantly and turning onion occasionally. Stir in cauliflower and garlic; add broth. Bring to boiling; reduce heat. Simmer, covered, for 25 minutes. Remove from heat. Stir in 1 tablespoon tarragon.
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. Return pureed mixture to pan.) Add half-and-half, the 1 teaspoon salt, and the white pepper. Heat through, stirring occasionally. Season to taste with additional salt.
If soup is too thick, thin with additional stock or broth. Ladle soup into bowls. Sprinkle with bacon and, if desired, additional tarragon.