Preheat oven to 350 degrees F. Line a large baking sheet with parchment paper.
In a food processor, pulse the butter with 1/2 cup of the powdered sugar until combined. Add the cornstarch, vanilla extract, and flour; process until a soft dough forms. Transfer the dough to a work surface and divide into 4 equal pieces.
On a lightly-floured work surface, roll each piece of dough into a 12-inch long rope. Cut each rope into 12 pieces and roll each piece of dough into a small ball. Transfer the balls to the prepared baking sheet. NOTE: These cookies do not spread very much.
Bake the cookies on the center rack of the oven for approximately 22 minutes or until the bottoms are golden, but the tops are still pale. Remove from oven.
While cookies are baking, place the remaining 1 cup sifted powdered sugar in a shallow dish.
While still warm, roll cookies in powdered sugar. Add 12 warm cookies to the bowl at a time and toss to coat.
While the cookies are still warm, poke 4 shallow holes into each cookie with a toothpick or skewer to make a buttonhole pattern. Let the cookies cool completely before serving or storing.
When cookies have cooled, roll them again in the confectioners sugar to give them a nice even coating of sugar. The cookies can be stored in an airtight container for up to 1 week.
Yields 4 dozen cookies.
Baby Button Cookie Recipe: https://whatscookingamerica.net/cookie/babybuttoncookies.htm