In a large bowl, sift together the cake flour, cornstarch, baking powder, and salt. Whisk to combine and set aside.

In the bowl of an electric stand mixer, beat together the softened butter, shortening, and granulated sugar on medium speed until very pale and fluffy, about 4 minutes. Scrape down the sides of the bowl as needed.

Add the egg and mix until combined. Then add the egg white, vanilla extract, and almond extract and mix until combined.

With the mixer running on low speed, gradually add the dry ingredients to the wet ingredients and mix just until combined. Do not overmix.

Transfer the dough to an airtight container and refrigerate for at least 2 hours. This step is crucial for the cookies to hold their shape.

About 10 minutes before the refrigeration time is up, preheat your oven to 375°F (190°C).

Scoop about 1/4 cup of chilled dough at a time and roll it into a ball. Place the dough ball on a lightly floured surface. Lightly sprinkle the top of the dough ball with flour. Using a flat, smooth object (like the bottom of a glass or a cookie cutter), press down evenly on the dough ball to flatten it to about 1/2 inch thick. The goal is to create a circular cookie that is about 2.5 to 3 inches in diameter.

Transfer the flattened dough to a baking sheet lined with parchment paper or a silicone baking mat. Repeat with the remaining dough, ensuring there is some space between cookies. You may need to use two baking sheets.

Bake for 9 to 11 minutes, or until the edges are lightly golden and the centers are set. Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.

In a large mixing bowl, combine the softened butter, powdered sugar, vanilla extract, and 2.5 tablespoons of the half-and-half. Use an electric hand mixer to beat until the frosting is smooth and fluffy. Add more half-and-half, one teaspoon at a time, if needed to reach desired consistency.

If using food coloring, add a few drops and mix until evenly colored.

Once the cookies are completely cool, frost them with the vanilla frosting. If adding sprinkles, apply them immediately after frosting each cookie, as the frosting begins to set quickly.