In the bowl of a stand mixer fitted with the paddle attachment, cream together the butter and sugar until light and fluffy.
In a separate bowl, whisk together the flour, baking powder and salt. In three increments, add the flour mixture to the creamed butter, mixing between each addition and scraping down the sides of the bowl as necessary. Add the milk and vanilla extract, blending until combined and the dough begins to come together in large pieces.
Use your hands to divide the dough in half, pressing it together to compact it into two disks. Wrap the disks securely in plastic wrap and refrigerate them until firm, about 1 hour.
Preheat the oven to 350ºF.
Once the dough has chilled, roll each disk out onto a lightly floured surface until it is 1/8-inch thick. Cut out as many cookies as possible with a doughnut-shaped cookie cutter. (See Kelly's Notes for cookie cutter alternatives.) Place the cut-out cookies on a Silpat or parchment paper-lined baking sheet and repeat with remaining dough.
Bake the cookies for 10 to 12 minutes, rotating the baking sheet half-way through, until the cookies are pale golden brown. Transfer the cookies to a wire wrack to cool completely.