Preheat the oven to 375°F. Lightly oil 2 baking sheets or line them with parchment paper.
In a medium bowl, sift or whisk together the flour with the baking powder and salt.
In another bowl, cream the sugar with the butter until light lemon-yellow, about 4 minutes. Beat in the egg until incorporated. Stir in the vanilla and milk, then add the flour mixture 1Ž2 cup at a time and mix until blended.
Wrap the dough in plastic wrap and refrigerate for at least 1 hour, or until firm.
On a lightly floured surface, roll the dough out until thin, about 1/8 inch. Cut into desired shapes, sprinkle with sugar, and place on the prepared baking sheets. Bake for 8 to 10 minutes, until browned. Transfer to wire rack to cool. Yields approximately 4 dozen cookies.