In a large bowl, whisk together 2 cups cake flour, ½ cup all-purpose flour, 1½ tsp baking powder, and ¼ tsp cream of tartar. Set the dry mixture aside.
In a second large mixing bowl, combine 1 cup sugar, ½ cup salted butter (softened), and ¼ cup butter-flavored shortening (softened). Using a hand mixer, beat on medium speed until the mixture is light and fluffy, about 2–3 minutes.
Add 3 tsp vanilla extract, then add the whole egg and the egg white. Beat on medium-low until the eggs are fully incorporated and the mixture is uniform. Turn off the mixer and scrape down the sides and bottom of the bowl with a rubber spatula.
With the mixer on low, gradually add the dry flour mixture to the wet ingredients. Continue mixing just until no streaks or pockets of flour remain and the dough is evenly combined.
Cover the bowl tightly with plastic wrap and refrigerate the dough overnight (or at least 8 hours).
When ready to bake, preheat the oven to 375°F. Line two baking sheets with parchment paper and set them aside. Keep the ⅓ cup sugar for dusting the tops of the cookies handy.
Remove the chilled dough from the refrigerator. Use a cookie scoop or tablespoon to portion dough into balls about 2–3 tablespoons each, depending on desired cookie size. Place the dough balls on the prepared baking sheets, spacing them at least 2 inches apart.
Flatten each dough ball to about ½ inch thick using the bottom of a cup or your hand. (Alternatively, you can roll the dough on a lightly floured surface to ½ inch thickness and cut circles with a round cutter.)
Bake one sheet at a time in the preheated oven for 10–12 minutes, or until the cookie edges are set and firm but not browned. Start checking at 10 minutes.
Immediately after removing the baking sheet from the oven, evenly sprinkle some of the reserved ⅓ cup sugar over the hot cookies. Let the cookies cool on the baking sheet for 2–3 minutes.
Transfer the cookies to a wire rack or cooling surface to cool completely. Repeat steps 9–11 for any remaining cookie sheets and dough, using the remaining sugar for topping.
Once cooled, serve and enjoy.