In a large mixing bowl, cream sugar, butter, and shortening until light and fluffy, about 2 minutes on medium-high. Add vanilla extract, almond extract and the whole egg. Mix, then add the egg white and 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 OR in the freezer for 45 minutes.
Preheat oven to 375 degrees. Line two baking sheets with parchment paper, set aside.
Using a large cookie scoop (#20- approximately 3½ 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 so that it is ½” thick and about 2½ wide. Place 6 cookies onto each prepared baking sheet. Place in oven and bake for 10-12 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, cream butter until it becomes light and smooth. Add salt, food coloring and 2 tablespoons of milk. With mixing speed on low gradually add powdered sugar. Once powdered sugar is incorporated, bring mixing speed up to high and beat for 2 minutes. Add more milk to reach the desired consistency. Frost onto cooled cookies with a spatula or a large star tip (to get the nest look). Add sprinkles, candies or coconut.