In a bowl of a stand mixer fitted with a paddle, mix butter and sugar on low speed to combine, then increase to medium speed and beat until light and creamy, about 3 minutes. Scrape down the sides of the bowl, and mix in the vanilla.
On the lowest speed, mix in the flour to combine then increase the speed to medium and beat until the dough begins to cling to the paddle and no longer looks dry. Do not wait for it to form a solid mass.
Transfer the dough to a board and use the heel of your hand to bring the dough together. Put the dough on a large piece of plastic wrap and pat into a rectangle ½ to ¾ inch thick. Refrigerate for 30 minutes.
Let the dough soften slightly, so it does not crack or break on the edges when rolling. Roll dough out between two sheets of parchment paper to about ¼ inch thick.
Using a 2 ½-inch round fluted cookie cutter cut out about 30 cookies and transfer to parchment paper lined sheet pan. Trimmings can be pressed together and re-rolled once for additional cookies. Refrigerate for 15 minutes to allow the dough to chill before baking.
Position the racks in the lower and upper thirds of the oven and preheat the oven to 350°F. Line two sheet pans with parchment paper.
Using a small offset spatula, transfer the chilled cookies to the sheet pans, leaving, at least, 1-inch of space between each cookie. Sprinkle the cookies with sugar.
Bake for 12 to 15 minutes, switching the position and rotating the sheets halfway through, until crisp and light golden brown. Remove from the oven and cool for 5 minutes on the sheet pan to firm the cookies up a bit.
Transfer cookies directly to a cooling rack to cool completely. Cookies can be stored in an airtight container for up to 10 days.
This dough can be refrigerated for up to 5 days or frozen up to 2 weeks; defrost frozen dough overnight in the refrigerator.