Combine first 8 ingredients with 3 tablespoons oil and 3/4 cup water in a blender; purée until smooth. Taste and adjust seasoning, if necessary. Cover and chill 1 hour.
Combine radish and next 3 ingredients in a small bowl. Add remaining 1 tablespoon olive oil, tossing to coat. Pour gazpacho into bowls, and top evenly with radish salsa. Garnish, if desired.