In a bowl of a standing mixer fitted with a paddle attachment, mix the butter and sugar until light and fluffy, about 5 minutes. Add the egg, milk and vanilla and beat on high speed until the egg is incorporated and the mixture is light and fluffy again. Add the flour and salt and mix on low speed until just incorporated. Wrap the dough in plastic wrap and chill in the refrigerator, preferably overnight or at least 3 hours.
On a lightly floured surface, roll the dough out into a rectangle that's ¼-inch thick. Use cookie cutters to cut out shapes. Using an offset spatula, transfer the cookies to a sheet pan lined with parchment paper. Collect scraps to roll out again, cut with cookie cutters and place on sheets. Place the cookies in the refrigerator for one hour to ensure they are well chilled before baking.
Preheat oven to 350° about 30 minutes before you're ready to bake the cookies. Place the sheet pans in the oven and bake the cookies until they're golden around the edges, about 15 to 18 minutes. Set aside to cool.
Make the icing: In the bowl of a standing mixer, whisk together the egg whites and confectioners' sugar until smooth. Stir in the lemon juice and vanilla.
Decorate the cookies with the royal icing. Serve immediately or store in an airtight container with a piece of white bread to keep the cookies soft.