In a mixing bowl, cream together the butter and powdered sugar.
Mix in the egg, vanilla, and zest. Thoroughly combine.
In another bowl, sift together the flour, baking soda, salt, and cream of tartar.
Add the dry ingredients in batches to the creamed mixture until you have combined everything well.
Cover and refrigerate the dough for at least 30 minutes. This is to harden up the fat so the cookies don't spread too much.
Preheat the oven to 350 degrees
Cut the dough in half.
Roll one half out onto a floured surface to around a quarter-inch thickness. Cut the cookies out with a round cookie cutter or what cookie cutter you like.
Transfer cookies to a parchment or silicone lined half sheet pan. Do no more than 8 cookies on a pan.
Bake for 10-14 minutes (depends on size of your cookie cutter) or until slightly browned.
Allow to cool on a wire rack before glazing.