Make the cookies. In a small mixing bowl, whisk together the flour, baking powder and cream of tartar. Set aside.

If using a food processor: In the bowl of a food processor fitted with the blade attachment, add the butter cubes and sugar. Turn the food processor to low speed until they are totally combined. You may have to spread the mixture around with a rubber spatula if gathering up and missing the blades. Add the egg and vanilla extract and pulse a few times until homogenous, running the rubber spatula along the sides and bottom to ensure everything is incorporated. Pour in the flour mixture and pulse until the dough comes together and begins to ball up. Dump the dough out onto a large piece of parchment paper that's been lightly dusted with flour. Place another piece of parchment on top and roll the dough out with a rolling pin it is 1/4-inch in thickness. You might have to split the dough in two to do this. Chill in the refrigerator for about 30 minutes or until the dough is firm and cold.

If using a stand mixer or hand mixer: In the bowl of a stand mixer fitted with the paddle attachment, or with a large mixing bowl and a hand mixer, beat the softened butter and sugar until combined. Beat in the egg and vanilla extract, scraping down the sides and bottom of the bowl halfway through, until everything is homogenous. Pour in the flour mixture and mix on low just until the dough begins to come together, the mixture is crumbly, and the last streaks of flour disappear. Form the dough into a disc, as flat as you can, and wrap it i plastic wrap. Chill for about 30 minutes before rolling out. Once chilled, roll the dough out with a rolling pin between two sheets of parchment paper until it is 1/4-inch in thickness. You might have to split the dough in two to do this. Chill in the refrigerator for about 30 minutes or until the dough is firm and cold.

Cut out the cookies. Remove the top piece of parchment and cut out as many shapes as you can. (Starting from the edges and then working in helps.) I recommend rolling out the scraps only 2 more times, ensuring that you chill and flour when the dough gets too warm and sticky.

Transfer as many cutouts as you can to a half sheet tray lined with parchment paper, leaving a little space between them (they won't spread.) Before baking, place them in the freezer (or fridge if that's not an option) until they are very cold and hard.

While the cookies are chilling, preheat the oven to 350°F.

Bake one tray at a time on the middle rack for about 8 or 9 minutes, rotating the tray halfway through — or until the cookies are set, dry, and just starting to get a little light brown color on the edges. (Smaller cookies and cookies with sharp appendages tend to cook and brown on the edges faster, so keep an eye on them.) If you prefer soft cookies, undershoot the timing by about 1 minute — and if you like very crisp cookies, leave them in for about a minute or two longer, while making sure you avoid over-browning or burning.

Allow the cookies to cool on the sheet tray for about 5 minutes and then carefully transfer them to a wire rack.

Make the buttercream. Using a hand mixer and a large mixing bowl, or a stand mixer with the paddle attachment, beat the butter until it's smooth. Add the powdered sugar and vanilla extract and beat on low speed until the mixture is creamy and thick. Pour in the melted and cooled butter and beat just until combined. Depending on how thick you want your buttercream, mix in 2 or 3 tablespoons of the whole milk.

Decorate the cookies using a small offset spatula, piping bags, a butter knife or even the back of a spoon. Finish with sprinkles and sanding sugars, if desired.

Allow the cookies to set for about two hours, uncovered (so the buttercream has a chance to crust over) before attempting to stack and store them in an air-tight container.