For the cookies: Whisk together the vanilla and egg in a small bowl and set aside.
Beat the butter with a handheld mixer in a large bowl until fluffy. Scrape down the sides of the bowl with a rubber spatula, add the granulated sugar, confectioners' sugar and salt and continue beating until light and fluffy, 2 to 3 minutes. Scrape the sides of the bowl again if needed, then add the egg mixture and beat for about 1 minute. Reduce the mixer to low speed and add the flour in batches, mixing just until blended.
Divide the dough in half and place each half between 2 pieces of waxed paper or lightly floured parchment. Use a rolling pin to flatten the dough into disks about 1/4 inch thick. Slide the covered disks onto a flat cookie sheet or the back of a rimmed baking sheet. Refrigerate until firm, about 2 hours or overnight.
Position the oven racks with even space all around and preheat the oven to 325 degrees F. Line two baking sheets with parchment paper.
Place 1 disk of the cookie dough on a clean work surface and peel off the top sheet of paper. Cut out cookies with a 1 1/2-inch round or decorative-shape cookie cutter directly on the parchment. Use a spatula to transfer the cookies to a prepared baking sheet, spacing them about 1-inch apart. Repeat with other disk of dough. Any excess dough can be re-rolled, refrigerated and cut.
Bake until the edges of the cookies are light golden, about 15 minutes. Transfer the cookies to a rack to cool.
For the icing: In a small bowl, whisk the confectioners' sugar with just enough milk to make a thick icing. Tint the icing with food coloring if desired. Transfer the 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 squeeze the icing onto the cookies as desired. Top with sprinkles, candies or colored sugar if using. Set aside for about 1 hour to let the icing harden.
Store in an airtight container for up to 1 week.
If desired, try one of these variations: Walnut-Orange: Add 3/4 teaspoon grated orange zest to the butter and 2/3 cup finely chopped walnuts to the flour. Pecan-Cinnamon: Mix 2/3 cup finely chopped pecans and 1 teaspoon ground cinnamon to the flour. Sprinkle the tops of the unbaked cookies 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 flour. If desired, brush the top of each unbaked cookie with some egg white and press a few un-toasted almond slices on the top.
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