In a medium bowl, combine cake flour, baking powder and salt and set aside.
Using an electric mixer, cream together butter and sugar until light and fluffy. Add in eggs one at a time stirring after each addition. Stir in lemon juice, lemon zest and vanilla until combined. Gradually add in flour mixture combined.
Chill dough for at least 1 hour in the refrigerator.
Preheat oven to 350ºF. Line a baking sheet with parchment paper.
Roll 3 tablespoons of dough into a ball (see note below for shaped cookies). Use flour on your hands if needed. Place ball on baking sheet and flatten to about 1/2 inch thick, molding sides to keep round shape if necessary. Continue with remaining dough, placing each 2 inches apart.
Bake in preheated oven for 10-12 minutes. Cool for a few minutes on baking sheet, and transfer to wire rack to cool completely before frosting.
For the frosting, using an electric mixer, combine butter and confectioner’s sugar, gradually adding the sugar. When all sugar is added, add in lemon zest, lemon juice and blend on medium until fluffy. Add in food color if desired. If frosting is too thick, add more lemon juice, water or milk until desired consistency is reached.
Note: For shapes, roll out chilled dough ¼ inch thick on a well floured surface. Flour rolling pin and cut out to prevent sticking. Cut out desired shapes and place on parchment-lined baking sheet. Bake in preheated oven for 10-12 minutes. Do not overbake. Cool for a few minutes and transfer to wire rack to cool.
Shapes will spread from the baking powder. To define the shape used after the cookie has cooled, place cookie cutter on top of cookie and firmly press all the way down, removing any excess cookie for the edges.
Note: When making these, I highly recommend refrigerating for much longer than an hour or to put them in the freezer for a shorter period of time. These were unbelievably sticky to deal with and I had to keep flouring my hands between every cookie.
Note 2: In my humble opinion, making each cookie with 3 tablespoons of batter is too much. The cookies end up huge and when I made a subsequent batch with leftover batter, they did great with 2 tablespoons.
Note 3: The frosting was too thick so I added 2 tablespoons of milk and the consistency was much better.