My mom has been making these longer than my sister and I have been alive. Because the dough has so much butter in it, it’s crucial to keep the dough cold. Pop it back in the fridge if it becomes too soft to handle.
In a large bowl, using an electric mixer on medium-high speed, beat the butter and sugar, stopping occasionally to scrape down the bowl, until fluffy and pale yellow, about 2 minutes. Beat in the egg and vanilla, stopping once or twice to scrape down the bowl, about 1 minute more. Add the flour and salt. Mix on low until just blended. Divide the dough into two disks and wrap in plastic. Refrigerate the dough until firm, 2 to 3 hours.
Arrange the racks in the top and bottom thirds of the oven; preheat to 350°. Roll out 1 disk of dough between 2 sheets of parchment paper to about 1/4 inch thick. Peel off the top sheet of parchment and, using cookie cutters, cut out the cookies. Divide between 2 parchment-lined baking sheets.
Bake the cookies, rotating and reversing the sheets halfway through, until pale golden, 10 to 12 minutes. Let cool on the sheets for about 5 minutes, then transfer to wire racks to cool completely. Repeat with the remaining dough on cooled, parchment-lined baking sheets.
Decorate the cookies with the Royal Icing. Let dry for at least 1 hour or overnight. Store in an airtight container, layering the cookies between sheets of parchment.