Whisk the egg and vanilla in a small bowl and set aside.
With a hand held mixer in a large bowl, cream the butter until fluffy. Scrape down the sides of the bowl with a rubber spatula, add both sugars, and continue beating until light, 2 to 3 minutes. Scrape the sides of the bowl again, if needed, then add the egg mixture, beating for about 1 minute. Sift the salt with the flour. Reduce the mixer to a low speed and then add the flour mixture, mixing just until blended.
Turn the dough out of the bowl. Divide the dough in half and place each half between 2 pieces of lightly floured parchment or waxed paper. Use a rolling pin to flatten the dough into a disk about 1/4-inch thick. Slide the dough/parchment sheet onto a flat cookie sheet, or on the back of a rimmed baking sheet. Refrigerate until firm, about 2 hours or overnight.
Evenly space the racks in the oven and preheat to 325 degrees F.
Place 1 disk of the cookie dough on a clean work surface and peel off the top sheet of paper.
Cut cookies with a 1 1/2-inch round or a decorative shaped cookie cutter directly on the parchment. Use a spatula to transfer the cookies to a parchment-lined or non-stick baking sheet, spacing them about 1-inch apart. Repeat with other sheet of dough. Any excess dough can be re-rolled, refrigerated, and cut.
Bake until the edges of the cookies are a light golden, about 15 minutes.
Transfer cookies to a rack to cool.
Whisk the confectioners' sugar with just enough milk to make a thick icing.
Transfer icing to a small plastic bag. Press the icing into a corner of the bag and snip off the corner with scissors to make a small opening. Gently press the icing onto the cookie to make a simple pattern. Decorate as desired. Set aside for about 1 hour to let the icing harden.
Store in an airtight container for up to 1 week.
Add 3/4 teaspoon grated zest of orange with the sugar, and 2/3 cup finely chopped walnuts with the flour.
Mix 1 teaspoon ground cinnamon with the flour, and add 2/3 cup finely chopped pecans with the flour.
Sprinkle the tops of the cookies before baking with a mixture of 2 tablespoons granulated sugar and 2 teaspoons ground cinnamon.
Almond: Reduce the vanilla extract to 1 teaspoon and add 1/2 teaspoon pure almond extract.
Add 3/4 cup chopped toasted sliced almonds to the dough with the flour. If desired, for decoration, brush the tops of the unbaked cookies with some egg white and press a few un-toasted almond slices onto the top of each cookie prior to baking.
