Whisk together cake flour, all-purpose flour, baking powder, cream of tartar, and salt in a mixing bowl and set aside.

In a stand mixer with paddle attachment, cream sugar, softened butter, and shortening until light and fluffy.

Add vanilla extract, almond extract, whole egg, and egg white to the mixer; blend until well combined. Scrape the sides with a rubber spatula.

Turn mixer to low and gradually add dry ingredients until no flour pockets remain.

Cover mixing bowl with plastic wrap and refrigerate dough for 2 to 3 hours.

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

Scoop 2 to 3 tablespoon dough balls onto baking sheets, flatten to ½ inch thickness using the bottom of a cup or roll and cut dough circles with a cookie cutter, placing six cookies per sheet.

Bake each sheet for 8 minutes or until edges are firm but not browned. Cool cookies on baking sheet 2 to 3 minutes, then transfer to a cooling rack.

Beat butter and shortening in a mixing bowl until light and fluffy using a hand mixer.

Add vanilla extract, almond extract, and salt and mix well.

Gradually beat in powdered sugar and milk one cup and tablespoon at a time until fully incorporated and smooth.

Add pink food coloring until desired color is reached.

Frost cooled cookies with a knife or piping bag and top with sprinkles.