Preheat oven to 350 degrees. Line 2 baking sheets with parchment paper; set aside.
COOKIES: In a medium mixing bowl, whisk together flour, baking soda, salt and cream of tartar until combined; set aside.
In the bowl of an electric stand mixer fitted with the paddle attachment, cream together sugar and butter until combined. Add 1 whole egg, egg yolk, vanilla and peppermint extracts; beat until combined. With mixer set on low speed slowly pour in dry ingredients, when the dough starts to come together, stop the mixer and scrape down the sides and bottom. Continue to beat until just combined.
Scoop out 1 1/2 tablespoons of dough at a time and roll into a balls. Transfer 8 dough balls to a prepared baking sheet and space 2-inches apart. Keep remaining dough balls chilled until ready to bake.
Bake for 10 to 11 minutes, cookies should appear slightly under-baked. Remove from the oven and allow to rest for several minutes on the baking sheet, transfer to a wire rack to cool completely.
WHITE CHOCOLATE: Place chocolate chips and shortening in a microwave safe bowl; Heat in the microwave on 50% power for 20 second intervals, stirring well between intervals, until the chocolate is melted and smooth.
Dip cooled cookies half way into white chocolate; allow excess chocolate to run back into the bowl, gently scraping the bottom of the cookie along edge of the bowl. Transfer cookies to wax or parchment paper and immediately sprinkle with peppermint bits.
Allow cookies to set at room temperature or in the refrigerator. Store cookies in an airtight container. Yields 30 cookies