To prepare cookies: Preheat oven to 350 degrees and line baking sheets with parchment paper.
In a large bowl, whisk together flour, baking soda, cream of tartar and salt, and reserve.
In a bowl of an electric mixer on medium-high speed, beat butter until creamy, about 1 minute. Add sugar and beat until light and fluffy, about 2 minutes. Add egg yolks, 1 at a time, beating well after each addition. Add vanilla extract and peppermint extract, and beat until well incorporated.
Reduce speed to low, add flour mixture and mix until just incorporated. Stir in candy pieces.
Drop tablespoon-sized cookies 2 inches apart on prepared baking sheets, and bake until tops are cracked and lightly browned, about 11 to 13 minutes. Remove from oven and cool 2 minutes before transferring cookies to a wire rack to cool completely.
To prepare icing: In a bowl of an electric mixer on medium speed, combine butter, cream cheese and peppermint extract, and beat until creamy. Reduce speed to low, add powdered sugar and salt, and mix until creamy. Add food coloring, 1 drop at a time, to reach desired color. Drizzle or spread icing on cookies.
