With a mixer, cream the butter and sugar together until light and fluffy. Gently stir in the eggs, vanilla, lemon zest and salt until well combined. Add half of the flour and stir until moistened. Add the remaining flour (the dough will be stiff). Form into a round disc, wrap in plastic and chill for 30 minutes. (You can do this a day or two ahead of time to make your life easier).
Preheat the oven to 375. Roll the dough out on a floured surface to about a ¼ - inch thick. Stamp out shapes using cookie cutters. Place them on an ungreased cookie sheet and bake for 8-9 minutes. Cool on a wire rack.
To make the icing, combine the sugar and half of the milk. Whisk until you have a thick icing, adding a few drops of milk if needed. Add food coloring to make different colors. Decorate. Let dry on a wire rack, then store in an airtight container.