In a large bowl combine butter, sugar, eggs, and vanilla extract. Beat using an electric mixer on high speed until light and fluffy.
In another bowl combine the flour, baking powder, baking soda, and salt. Gradually stir this dry mixture into the butter mixture until well blended.
Cover bowl with plastic wrap and chill for 2 hours. Do not skip this!
Preheat oven to 400°F. Line cookie sheets with parchment paper (do not grease cookie sheets, and use parchment paper ONLY).
On a very lightly floured surface roll out the dough into about 1/4-inch thickness. Cut into desired shapes using cookie cutters. Place cookies 2-inches apart on cookie sheet.
Bake 4-6 minutes. Remove cookies to wire racks to cool completely before icing.
For the frosting: in a small bowl mix the confectioners sugar with milk (start with 1-2 tablespoons, but you will likely need more milk for the perfect spreading consistency).
Whisk in the corn syrup and vanilla extract until the icing is smooth and glossy. If the icing is too thick add in a small amount more of corn syrup.
Divide into as many separate bowls as you wish for different colors. Add in food coloring until desired intensity is achieved, always starting with a smaller amount and adding gradually.
Paint the icing over the cookies using a brush, or dip tops of cookies into icing. Allow to set on waxed paper.