Beat butter and white sugar together in the bowl of a stand mixer using the paddle attachment until light and fluffy, about 3 minutes. Add egg and vanilla extract; beat 1 minute more. Slowly add flour mixture and blend until dough begins to pull away from the sides of the bowl.
Turn dough onto a piece of parchment paper. Knead dough a few times using your hands and form into 2 discs. Wrap each dough disc in plastic wrap and refrigerate for 2 hours.
Preheat oven to 350 degrees F (175 degrees C). Line a baking sheet with a silicone mat.
Remove 1 disc of dough from the refrigerator. Place dough on a lightly floured work surface; top with a piece of parchment paper. Roll out dough to about 1/3-inch thickness. Cut dough with round cookie cutters and transfer to the prepared baking sheet.
Place baking sheet with cookies on it in the freezer or refrigerator for 3 to 5 minutes.
Bake in the preheated oven until cookies are just barely beginning to turn golden brown, 8 to 10 minutes. Cool cookies on the baking sheet for 1 minute; transfer to a wire rack to cool completely. Repeat with second disc of dough.
Wipe down all equipment for making the icing with vinegar, including spatulas, measuring cups, and bowls.
Combine confectioners' sugar and meringue powder in the bowl of a stand mixer using the paddle attachment.
Mix water, corn syrup, and clear vanilla extract together in a separate bowl. Pour water mixture into sugar mixture and beat on medium speed until stiff peaks form in the icing, 4 to 5 minutes.
