Position one oven rack in the top third of the oven and another in the bottom third. Heat oven to 350ºF. Line 2 half-sheet pans with parchment paper and set aside.
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 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 the prepared half-sheet pans, approximately 35 cookies per pan. Use the heel of your hand to slightly flatten each ball. Place both pans in the oven and bake, rotating the pans halfway through, until golden brown, 15 to 20 minutes. Remove the pans to a wire rack to cool completely before removing the cookies from the pan.