In the bowl of a stand-mixer with the paddle attachment, or in a large bowl by hand, mix the butter and sugar until combined and creamy.

Add the eggs and mix until evenly combined

Mix in the flour, baking soda, and cream of tartar until fully incorporate and no clumps of flour is visible. 

Stir in the milk and vanilla, using your hands to bring the dough together in the end.  Wrap the dough into two large disks, wrap in clingfilm, and refrigerate for 30 minutes.

Preheat your oven to 350° F and line two baking trays with  parchment paper.

Sprinkle flour over a work surface and place one disk of dough in the center.  Dust flour over a rolling pin and roll the cookie dough to about ¼-½ inch thick.  Use your favorite cookie cutters to cut shapes out of the dough and place on the baking tray.  Re-roll the dough and repeat the same process until all of the dough is used

Bake the cookies for 10-12 minutes, until just golden around the bottom.  Let the cookies cool on the tray for 5 minutes then transfer to a wire rack to cool completely.

While the cookies cool, make the frosting.  In the bowl of a stand mixer with the whisk attachment or in a large bowl  with a handheld mixer, add the powdered sugar, vanilla, and 1 tablespoon of water.  Whisk until combined, adding more water ½ tablespoon at a time until the desired consistency is reached.

Once the cookies are cooled, pipe or drizzle the frosting over top, decorating however you please!