Line two baking sheets with parchment paper.
Cream the butter and sugar in a large bowl until smooth.
Add the egg and vanilla and mix until combined.
Whisk the flour, baking powder, and salt in a separate bowl.
Add the dry ingredients to the butter mixture.
Mix until a soft dough forms.
Divide the dough in half and shape each half into a flat disc.
Wrap each disc tightly.
Chill the dough for 1 hour.
Preheat the oven to 350 degrees Fahrenheit.
Roll out one disc between parchment sheets.
Cut shapes using cookie cutters.
Place the cut shapes onto the lined baking sheets.
Repeat with the second disc.
Bake the cookies for 8 to 10 minutes.
Remove the cookies when the edges look set but still pale.
Cool the cookies on the baking sheet for 5 minutes.
Transfer the cookies to a rack to finish cooling.
Make the icing by mixing the powdered sugar, milk, and vanilla.
Add more milk for thinner icing or more powdered sugar for thicker icing.
Color the icing as desired.
Decorate the cooled cookies.
Let the icing set before storing.