In the bowl of a stand mixer or a large mixing bowl with a hand mixer, cream sugar, butter, and shortening until light and fluffy, about 2 minutes on medium-high. Add egg, egg white and vanilla extract, mix until combined.
In a small mixing bowl whisk together cake flour, all-purpose flour, baking powder, cream of tartar and salt. With mixing speed on low, gradually add dry ingredients until no flour pockets remain. Place dough in an air-tight container and refrigerate for at least 3 hours.
Preheat oven to 375 degrees. Line two baking sheets with parchment paper or a silicone baking mat, set aside.
Using a large cookie scoop (#20- approximately 3 1/2 tbsp.), create balls of dough, lightly roll on a floured surface so that it covers the ball of dough completely. Press each ball of dough flat with the bottom of a glass so that it is 1/2" thick and about 2 1/2 wide. Place 6 cookies onto each prepared baking sheet. Place in oven and bake for 9-11 minutes (the cookies will still look soft in the centers, do not over bake) or until the cookies become firm on the edges. Remove and allow to cool for 3 minutes before transferring to a wire rack to cool completely.
Meanwhile, to make the frosting, in the bowl of a stand mixer fitted with the whisk attachment, or a large mixing bowl with a hand mixer, cream butter until it becomes light and smooth, about 2 minutes. Add peppermint extract, vanilla extract, salt, and food coloring. Beat for 1 minute. With mixing speed on low gradually add the powdered sugar. Once the powdered sugar is incorporated, add milk and bring mixing speed up to high. Beat for 2 minutes. Add more milk to reach the desired consistency.
Frost onto cooled cookies and top with desired decorations.