Preheat the oven to 350 degrees and line a baking sheet with a baking mat or parchment paper.
Combine the sugar and butter in a large bowl and beat until light and fluffy, about 2 minutes. Add the vanilla and combine. Add the flour and cornstarch and mix on low speed until well combined.
Cover the cookie dough and refrigerate for 30 minutes.
Shape the dough into round 1-inch balls and place a few inches apart on the cookie sheet. Bake for 10-12 minutes or until the edges are lightly browned. Let the cookies cool on the baking sheet for one minute and then remove and let them cool on a wire rack.
For the glaze, combine the powdered sugar, butter, vanilla extract and enough milk for desired glazing consistency in small bowl. Drizzle or spread the glaze over cooled cookies.
Enjoy and store any extras in an air-tight container.
Notes
This is the plainest version of this cookies but feel free to change it up! You can easily add food coloring to the dough and or icing. It's easy to sprinkle on sprinkles, crushed candy canes, and sprinkling sugars.