1. Place butter and sugar in a large bowl; beat with a mixer at medium speed for 5 minutes or until light and fluffy. Add egg, 1 tablespoon lime rind, and lime juice; beat 2 minutes or until well combined.
2. Weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine flour, baking powder, and 1/8 teaspoon table salt; stir with a whisk. Add flour mixture to butter mixture, and beat just until combined.
3. Divide dough into 2 equal portions. Shape each portion into a 6-inch log. Wrap logs individually in plastic wrap; chill 3 hours or until firm.
4. Preheat oven to 350°.
5. Cut each log into 16 (about 1/3-inch-thick) slices, and place 1 inch apart on baking sheets lined with parchment paper. Sprinkle the cookies evenly with remaining 2 teaspoons lime rind, turbinado sugar, and sea salt. Bake at 350° for 10 minutes or until edges are lightly browned. Cool for 2 minutes on pans on a wire rack. Remove from baking sheets, and cool completely on wire rack.
6. Place vanilla ice cream and sherbet in a medium bowl; lightly fold and swirl together. Scoop 1/4 cup ice cream mixture onto bottom of one cookie, and top with one cookie. Cover each sandwich with plastic wrap; freeze 4 hours or until firm.