In a bowl of a stand mixer fitted with a paddle attachment, mix butter and sugar at a low speed to combine, then increase to medium speed 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 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 gently knead it together. Put the dough on a large piece of plastic wrap and shape it into a rectangle ½ to ¾" thick. Cover and refrigerate for 30 minutes.

Let the dough soften slightly, so it does not crack or break on the edges when rolling, for 5 to 10 minutes. Then roll out between two sheets of parchment paper to about ¼" thick.

Using a 2 ½" round fluted cookie cutter, cut out about 30 cookies and transfer them to two parchment paper-lined sheet pans, leaving 1" space between them. Refrigerate for 15 minutes to allow the dough to chill before baking. Sprinkle the cookies with sugar.

Set the oven racks to the lower-middle and upper-middle position. Preheat to 350°F (177ºC).

Bake for 12 to 15 minutes, switching the position and rotating the sheets halfway through. Remove from the oven and cool for 5 minutes on the sheet pan to firm the cookies up.

Transfer cookies to a cooling rack to cool completely.