Preheat the oven to 350°F (175°C). Line baking sheets with parchment paper.

In a large bowl, cream together the butter and sugar until light and fluffy. Beat in the egg and vanilla extract.

Gradually mix in the flour and salt until a dough forms. If the dough is too sticky, chill it for 30 minutes.

Roll out the dough on a floured surface to about 1/4-inch thickness. Use star-shaped cookie cutters to cut out cookies.

Place the cookies on the prepared baking sheets and bake for 8-10 minutes, or until the edges are lightly golden.

Let the cookies cool completely on a wire rack before decorating.

For the icing, mix powdered sugar and milk until smooth. Divide into two bowls and tint one portion with blue food coloring.

Decorate the cookies with the white and blue icing, creating festive Hanukkah designs. Let the icing set before serving.