Stir together shredded cucumber, chicken broth, sour cream, snipped dillweed, lemon juice, and pepper in a medium mixing bowl.
Pour mixture into a nonmetal freezer container. Cover and freeze for several hours or until almost firm. Break mixture into small chunks; transfer to a chilled bowl. Beat with an electric mixer until smooth but not melted. Return to container. Cover and freeze for at least 6 hours or until firm.
To serve, transfer sorbet to a serving bowl. If desired, garnish with a fresh dill sprig. Makes about 2-1/2 cups (5 servings).