Preheat oven to 350 degrees F (175 degrees C). Line a baking sheet with parchment paper.
Beat 1/2 cup butter in a bowl with an electric mixer until smooth. Add white sugar, baking powder, and baking soda; beat until blended. Beat in 1/3 cup pink lemonade concentrate and egg. Beat in flour until dough comes together.
Drop dough onto the prepared baking sheet by rounded spoonfuls, spacing them 2 inches apart.
Bake in the preheated oven until edges are golden, 10 to 12 minutes. Transfer to a wire rack to cool, about 10 minutes.
Combine confectioners' sugar, 2 tablespoons pink lemonade concentrate, and 2 tablespoons butter in a large bowl; beat with an electric mixer until icing is smooth. Beat in red food coloring.