Sift together the cake flour, cornstarch, baking powder, and salt into a large bowl and whisk a few times to combine; set aside.
In the bowl of a stand mixer fitted with the paddle (or using a hand mixer), beat the 1/2 cup butter, 1/4 cup shortening, and 1 cup granulated sugar on medium speed until very pale and fluffy, about 3–4 minutes, scraping the bowl as needed.
Add the whole egg and beat until combined, then add the egg white, 2 tsp vanilla, and 1/2 tsp almond extract and beat until incorporated.
With the mixer on low, slowly add the dry ingredients and mix just until a soft dough forms; do not overmix.
Transfer the dough to an airtight container and refrigerate for 2 hours. Preheat the oven to 375°F (190°C) during the last 10 minutes of refrigeration.
Scoop scant 1/4 cup portions of dough and roll each into a ball. On a lightly floured surface, lightly dust the top of a ball with flour, then press and evenly flatten each ball to about 1/2 inch thick using a flat, smooth object so the cookie is roughly 2 2/3–2 3/4 inches across.
Place flattened cookies on a Silpat- or parchment-lined baking sheet, spacing them a few inches apart. Repeat with remaining dough, using a second baking sheet as needed.
Bake in the preheated oven for 9–11 minutes until edges are set but centers remain pale. Allow cookies to cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
To make the vanilla frosting: in a large bowl, beat 1/3 cup butter until smooth, then add 2 1/2 cups powdered sugar and 1/2 tsp vanilla and beat until combined. Add 2 1/2–3 tbsp half-and-half as needed to reach a smooth, fluffy spreading consistency. Add food coloring if desired.
Frost cooled cookies with the vanilla frosting. If adding sprinkles, apply them immediately after frosting each cookie so they stick before the frosting begins to set.