1. In a large bowl, combine butter, granulated sugar, and lemonade mix. Beat with a mixer on medium-high speed until fluffy, about 3 minutes. Add egg and food coloring; beat well. Gradually add flour and salt; beat until dough begins to cling.
2. Gather dough into a ball and shape into a 7x2 1/2-inch log. Wrap in waxed paper. Chill 3 hours so it's firm enough to slice.
3. Preheat oven to 350 degrees F. Line cookie sheets with parchment paper. Place decorating sugar in a shallow bowl. Remove waxed paper from dough. Brush outside of the dough log with corn syrup. Cut log crosswise into 1/8-inch thick slices. Lightly roll edges of the slices in the pink sugar. Place on cookie sheets about 1 inch apart. Cut some cookies in half, if desired. Score the top of cookies with the back of a small knife to make them resemble citrus segments.
4. Bake about 8 minutes, or until edges are just firm. Let cool on a wire rack.
5. Mix powdered sugar and lemon juice; place in a resealable plastic bag. Snip a small hole in one corner of the bag. Pipe icing onto scored lines and add dots. Let stand until icing is set.