In a stand mixer with the paddle attachment, cream together butter, shortening and sugar until fluffy.
Add whole egg, egg white, vanilla extract, and peppermint extract. Mix until combined.
Sift in dry ingredients and mix until just combined.
Transfer dough to an airtight container and refrigerate at least 2 hours.
Once ready to bake, preheat oven to 375 degrees F.
Scoop 1 tbsp of dough and roll into a ball.
On a lightly floured surface, flatten dough ball into a ½-inch thick disc. Place on a cookies sheet lined with parchment paper or a silicone baking mat.
Bake 9-12 minutes. Bottom edges should not be browned. Remove from oven and cool 1-2 minutes on the pan before removing to a wire rack to cool completely.
Frosting
In a bowl, cream together cream cheese and butter until fluffy. Add peppermint extract. Slowing mix in powdered sugar. If the frosting is too thick, add milk a teaspoon at a time.
Frost completely cooled cookies. Sprinkle crushed peppermints on top of each cookie.