In a large mixing bowl, whisk together the cake flour, all-purpose flour, baking powder, and cream of tartar. Set aside.

In another large mixing bowl, beat together the sugar, butter, and shortening with a hand mixer until light and fluffy, about 2-3 minutes.

Add vanilla extract and the whole egg to the butter mixture. Beat until combined. Add egg white and continue beating until well combined. Use a rubber spatula to scrape the sides of the bowl.

Turn the hand mixer on low and gradually add the dry ingredients to the wet ingredients. Mix until no flour pockets remain, being careful not to overmix.

Cover the mixing bowl with plastic wrap and refrigerate for 2-3 hours. This step is essential for preventing spreading.

Preheat oven to 375°F and line 2 baking sheets with parchment paper.

Remove cookie dough from refrigerator and roll out on a floured surface to about ½ inch thick. Cut out desired shapes with floured cookie cutters.

Place 6 cookies on each prepared baking sheet. Bake one sheet at a time for 10-12 minutes, until edges are firm without browning.

Let cookies cool on the baking sheet for 2-3 minutes, then transfer to a wire rack to cool completely before frosting.

For the frosting, beat together 1 cup softened butter and 4 cups powdered sugar in a medium bowl. Add vanilla extract. With the mixer running, drizzle in heavy cream until frosting reaches desired consistency. Beat until fluffy.

Spread frosting onto cooled cookies and decorate with sprinkles immediately, before the frosting sets.