Position 1 oven rack in the top third of the oven and another in the bottom third. Heat the oven to 350 degrees F.
Sift together the flour, baking powder, and salt in a small bowl and set aside. Cream the butter and vanilla sugar in the bowl of a stand mixer on medium speed for 2 minutes, stopping to scrape down the sides of the bowl after 1 minute. Add the egg and incorporate on medium speed for 30 seconds. Scrape down the sides of the mixer bowl. Add the vanilla extract and milk and blend on low speed for 15 seconds. Add the flour mixture and mix on low speed just to incorporate. Chill the batter in the refrigerator for at least 10 minutes before scooping.
Scoop the batter in teaspoon-sized balls and arrange them on 2 parchment paper-lined half sheet pans, approximately 35 cookies per pan. Use the heel of your hand to slightly flatten each ball. Bake, 2 pans at a time, rotating the pans halfway through the baking, until golden brown, about 15 to 20 minutes. Remove the pans to a cooling rack to cool completely before removing the cookies from the pan.