. Prepare the Cookies:

1. Preheat Oven: Preheat your oven to 350°F (175°C). Line baking sheets with parchment paper or silicone mats.
2. Mix Dry Ingredients: In a medium bowl, whisk together the cake flour, cornstarch, baking powder, and salt. Set aside.
3. Cream Butter and Sugar: In a large bowl, cream the butter, shortening, and granulated sugar until light and fluffy, about 2-3 minutes.
4. Add Wet Ingredients: Beat in the egg and vanilla extract until fully combined. If desired, add a small amount of pink gel food coloring and mix until the dough is evenly colored.
5. Combine Dry and Wet: Gradually add the dry ingredients to the wet mixture, mixing until a soft dough forms.
6. Shape the Cookies: Roll the dough into 1-inch balls and place them on the prepared baking sheets, spacing them about 2 inches apart. Flatten slightly with the palm of your hand or the bottom of a glass.

2. Bake the Cookies:

1. Bake for 9-11 minutes, or until the edges are just set. The cookies should remain soft and light in color.
2. Remove from the oven and allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.

3. Make the Strawberry Frosting:

1. Blend Ingredients: In a medium bowl, beat the softened butter until creamy. Add the freeze-dried strawberry powder, fresh strawberry puree, lime or lemon juice, and vanilla extract. Mix until smooth.
2. Add Sugar Gradually: Gradually add the powdered sugar, beating until the frosting is light and fluffy. Adjust the consistency with additional powdered sugar or a splash of milk if needed.

4. Frost the Cookies:

1. Spread or pipe the frosting onto the cooled cookies. For a decorative touch, sprinkle with extra freeze-dried strawberry crumbs or sanding sugar.
2. Allow the frosting to set slightly before serving.