In a large mixing bowl add the cake flour, all-purpose flour, baking powder, and cream of tartar. Whisk together, then set aside.
In a large mixing bowl, add the sugar, butter, and shortening. Use a hand mixer to beat together until light and fluffy. Add vanilla extract, and the whole egg. Beat together until combined. Add the egg white and continue beating until well combined. Turn off the mixer and use a rubber spatula to scrape the sides of the mixing bowl.
Turn the hand mixer on low and gradually add dry ingredients to wet ingredients. Mix until no flour pockets remain. Cover the mixing bowl with saran wrap and place it in the refrigerator overnight.
Preheat the oven to 375°F and line 2 baking sheets with parchment paper. Set aside.
Remove cookie dough from the refrigerator and roll the dough out on a flat surface to about ½ inch thick. Cut out desired shapes with floured cookie cutters. Place 6 cookies on one baking sheet and 6 on another.
Place one cookie sheet in the oven at a time and cook for 10 to 12 minutes. You want the edges to be firm without turning brown. Let the cookies cool for 2 to 3 minutes on the baking sheet. Place the other baking sheet in the oven and cook.
In a medium mixing bowl, beat together 1 cup butter and 4 cups powdered sugar. With the beater turned on, drizzle heavy cream into butter/sugar mixture and beat until fluffy. Add more cream to thin out frosting to a desired spreadable consistency. Spread frosting onto cooled cookies and decorate with sprinkles.