Begin by placing butter and sugar in a large mixing bowl or stand mixer fitted with the paddle attachment. Cream them together until the mixture is light and fluffy, which usually takes about 1 to 2 minutes. This step is crucial, as it incorporates air into the mix and ensures a nice texture.
Add in the egg, vanilla, salt, baking powder, and cocoa powder. Mix until everything is well combined. You should see a rich chocolate color forming.
Gradually mix in the flour until the dough is smooth and well-formed. The dough should be somewhat soft but not sticky.
Take half of the cookie dough and lay it on a sheet of wax or parchment paper. Press it into a disk shape, then cover with another sheet of paper. Roll it out to your desired thickness of between ¼ to ½ inch.
Repeat the rolling process with the second half of the dough. Once both portions are rolled out, place them in the refrigerator to chill for at least 30 minutes. This chilling step is important as it helps the cookies hold their shape during baking.
Preheat your oven to 350°F (175°C) while waiting. Prepare your baking sheets by lining them with parchment paper or silicone baking mats.
Remove one portion of chilled dough from the refrigerator. Use cookie cutters to cut out desired shapes and place them on the prepared baking sheets, ensuring they are spaced about 2 inches apart. Chill again for 15 minutes before baking.
Bake your cookies in the preheated oven until they are no longer glossy, which typically takes about 10 to 15 minutes depending on their thickness and size. Keep an eye on them to avoid overbaking!
Once the cookies are baked, allow them to cool completely on a wire rack before moving on to the frosting.
To make the frosting, beat the remaining butter and powdered sugar in a stand mixer until it becomes crumbly. Add in the salt and vanilla, then mix in 3 tablespoons of milk. Keep adding milk, one teaspoon at a time, until you achieve a smooth, piping consistency.
Frost your cooled cookies as desired, letting them sit at room temperature to crust before stacking. You can decorate with food coloring or sprinkles for that extra festive flair.