Prepare the dough: Begin by preheating your oven to 350°F (175°C). In a mixing bowl, whisk together the flour, baking soda, salt, and cream of tartar, setting the mixture aside for later.

Cream the wet ingredients: Using an electric mixer fitted with the paddle attachment, combine the softened butter and granulated sugar, creaming them together until smooth. Add the egg and egg yolk, followed by the vanilla and peppermint extracts, mixing until fully incorporated.

Combine dry and wet ingredients: Gradually add the dry flour mixture into the wet ingredients, mixing on low speed. The dough will be quite thick, so if needed, stop the mixer to incorporate any flour that has settled at the bottom by mixing it by hand.

Shape the cookies: Scoop about 1 1/2 tablespoons of dough and roll it into a ball. Place each dough ball onto a baking sheet lined with parchment paper or a silicone mat, leaving 2-inch spaces between them. If you’re not ready to bake all the dough, chill the unbaked dough balls in the refrigerator until ready.

Bake the cookies: Bake the cookies for 10 to 11 minutes, until they appear slightly underbaked in the center. Remove them from the oven and let them rest on the baking sheet for a few minutes before transferring them to a wire rack to cool completely.

Prepare the white chocolate dip: In a microwave-safe bowl, combine the white chocolate chips and shortening. Microwave in 20-second intervals at 50% power, stirring after each interval, until the mixture is completely melted and smooth.

Dip and decorate: Once the cookies have cooled, dip half of each cookie into the melted white chocolate, letting the excess drip off. Place the dipped cookies on a sheet of wax paper and immediately sprinkle with peppermint bits or crushed candy canes for a festive touch.

Let set: Allow the white chocolate to harden at room temperature or in the refrigerator before serving. Store the finished cookies in an airtight container at room temperature to preserve their freshness.