In a medium bowl, cream together the butter and sugar with an electric mixer until light and fluffy.
In a separate bowl, whisk together the flour, baking powder and salt. In 3 increments, add the flour mixture to the creamed butter, making sure that all the batter is incorporated. Add the milk and vanilla extract, blending until combined. The dough should come together in large pieces.
Divide the dough in half, pressing it together into 2 disks. Wrap the dough disks in plastic wrap and refrigerate them until firm, about 1 hour.
Preheat the oven to 350 degrees.
Once the dough has chilled, roll each disk out onto a lightly floured surface until it is 1/8-inch thick. Cut out cookies using two round cookie cutters: one 2-inch, and the other smaller to form donut-like shapes. Place the cutout cookies on parchment paper-lined baking sheet and repeat with remaining dough.
Bake for 10 minutes, rotating the baking sheet at 5-minute mark until the cookies are pale golden brown. Transfer the cookies to a wire rack to cool completely.