Whisk the flour, cocoa powder, baking soda, cream of tartar, and salt in a medium bowl until evenly combined, with no visible streaks of cocoa. Set aside.

Beat the butter on medium-high speed until smooth and creamy, about 1 minute. Add the powdered sugar and mix on low speed until mostly combined, scraping down the sides to prevent dry spots.

Increase the speed to medium-high and mix in the egg and vanilla extract until the dough looks smooth and slightly fluffy.

Lower the mixer speed and gradually add the dry ingredients. Mix just until the dough pulls away from the sides and forms a soft ball. If the dough feels sticky, add flour 1 tablespoon at a time.

Divide the dough into two portions. Roll each into a ¼-inch thick slab between sheets of parchment paper. Stack the slabs on a baking sheet and refrigerate for at least 1 hour, preferably overnight.

Preheat the oven to 350°F and position the rack just above center. Line baking sheets with parchment paper.

Lightly flour the cookie cutters and cut out shapes. Place cookies 1 to 1½ inches apart on the prepared sheets. Bake for 7 to 8 minutes, until the edges are set but the centers remain soft.

Let the cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack and cool completely before frosting.

Beat the butter on medium-high speed for 1 to 2 minutes until smooth and creamy.

With the mixer on low speed, alternate adding the powdered sugar and heavy cream until fully incorporated. Adjust the consistency with additional powdered sugar if a thicker frosting is needed.

Add the vanilla extract and salt, then beat on medium-high speed until light and fluffy. Tint if desired and frost the cooled cookies.