In a large mixing bowl, combine the softened butter, butter-flavored shortening, and sugar. Beat on medium speed until light and fluffy, about 2-3 minutes.
Add the whole egg, egg white, and vanilla extract to the butter mixture. Beat until well combined.
In another bowl, whisk together the cake flour, all-purpose flour, baking powder, and cream of tartar. Gradually add the dry mixture to the wet ingredients, mixing on low speed until just combined.
Cover the dough with plastic wrap and refrigerate for at least 1 hour.
Preheat your oven to 350°F (175°C) while the dough chills.
Once the dough is chilled, turn it out onto a lightly floured surface. Roll it out to about ¼-inch thickness. Use cookie cutters to cut out your desired shapes. Place the cookies on a baking sheet lined with parchment paper.
Sprinkle the tops of the cookies with sugar before baking.
Bake the cookies in the preheated oven for 8-10 minutes, or until the edges are just starting to turn golden.
Remove the cookies from the oven and let them cool on the baking sheet for a few minutes before transferring them to a cooling rack. Allow them to cool completely before decorating.