Adjust an oven rack to the middle of the oven and preheat to 350°F [180°C]. Line three sheet pans with aluminum foil or parchment paper.

In a small bowl, whisk together the flour, baking soda, salt, and cream of tartar.

In the bowl of a stand mixer fitted with a paddle, beat the butter on medium speed until creamy, about 1 minute. Add the granulated sugar and beat until light and fluffy, 2 to 3 minutes. Add the egg and mix on low speed until combined.

Add the cocoa powder, red food coloring, and vanilla, and mix on low until combined. Scrape down the sides as needed.

Add the flour mixture and mix on low speed until completely combined.

Form the dough into 3 oz [85 g] balls [1/4 cup] and place 4 cookies an equal distance apart on the sheet pans.

Bake the cookies one pan at a time until the dough balls have spread but are puffed slightly in the center, about 8 minutes.

Lift one side of the sheet pan up about 4 inches [10 cm] and gently let it drop to set the edges of the cookies. After 2 minutes, repeat lifting and dropping the pan. Continue to repeat this a few more times to create ridges around the cookies.

Bake for a total of 13 to 16 minutes, until the edges are set and golden but the centers are not fully cooked.

Transfer the pan to a wire rack and let the cookies cool for 10 minutes before moving them to a wire rack to cool completely.

For the icing, melt the butter in a medium bowl. Add the cream cheese, 2 tablespoons of water, vanilla, and salt, and whisk to combine.

Add 1 cup [120 g] of confectioners' sugar and mix with a spatula. Add more sugar as needed to reach desired consistency. Adjust thickness with more water if necessary.

To assemble, dip half of a cooled cookie into the icing and decorate with holly sprinkles if desired. Let the icing set before serving.