Preheat oven to 350 degrees F. Line 2 baking sheets with parchment paper.
In a medium bowl, combine flour, baking soda, cream of tartar, and salt, and whisk to combine.
In a large bowl with electric mixer, beat sugar and butter until light and creamy, about 2 minutes.
Add egg and egg yolk, one at a time, beating well between each addition.
Add vanilla and peppermint extracts and beat to combine.
Add dry mixture to wet ingredients and beat until well combined.
Scoop out dough, about 1 ½ to 2 tablespoons at a time, and shape into cookie dough balls.
Place on prepared baking sheets.
Bake in preheated oven for 10-12 minutes, until edges are set but the middles appear slightly underbaked. They will set up!
Let cool for 5 minutes on pan then transfer to cooling rack to cool completely.
Once cookies are cooled, combine white chocolate and shortening in a microwave safe bowl.
Microwave in 30 second intervals, stirring well between each interval, until white chocolate is melted.
Dip each cookie in the white chocolate and then sprinkle with crushed candy canes.
Let cool completely at room temperature or in the refrigerator.
Store leftovers in an airtight container at room temperature.