FOR THE COOKIES: In a mixing bowl, whisk together cake flour, cornstarch, baking powder and salt for 30 seconds. Add 1 cup freeze dried strawberries to a food processor and pulse until ground into a fine powder. Sift ground strawberries through a fine mesh strainer (to remove seeds and the few small pieces remaining) into the flour mixture and whisk to combine.
In the bowl of an electric stand mixer fitted with the paddle attachment, whip butter, shortening and sugar on medium speed until very pale and fluffy, about 4 minutes. Mix in egg, then mix in egg white and vanilla.
Blend in food coloring if desired (I added 1 toothpick swish, dipped in gel). With mixer set on low speed, slowly add in dry ingredients and mix just until combined. Transfer dough to an airtight container and refrigerate 2 hours. Preheat oven to 375 during the last 10 minutes of refrigeration.
Scoop dough out about 3 Tbsp (50 g each) at a time and roll into a ball. Place dough ball on a lightly floured surface, sprinkle top lightly with flour, then using something flat and smooth (such as the storage container lid), press and evenly flatten dough to about 1/3-inch thick (so you should have about a 2 3/4 " circle).
Transfer flattened dough to parchment paper lined baking sheets and repeat process with remaining dough. Bake in preheated oven 8 - 9 minutes. Allow to cool 2 minutes on baking sheet before transferring to a wire rack to cool. Cool completely then frost with Strawberry Frosting.
FOR THE FROSTING: In the bowl of an electric stand mixer, fitted with the paddle attachment, whip butter until pale and fluffy. Meanwhile, pulse 1/3 cup freeze dried strawberries in the food processor until finely ground.
Using a fine mesh strainer to remove seeds, sift freeze dried strawberries over butter and mix to combined. Stir in 3 Tbsp fresh strawberry puree, lime or lemon juice and vanilla extract (mixture will look separated and that's okay). Slowly mix in powdered sugar and whip until well combined.