In your food processor, pulse the flour, powdered sugar and salt until combined.With the processor running, drop in the pieces of butter, one at at time, pulsing a few times until the butter is evenly distributed, and the dough is moistened and crumbly.Add the vanilla and the milk, then process until the dough starts to stick together.

Dump the dough out onto a clean counter and use your hands to finish bringing the dough together, kneading in any stray bits of flour. The dough should be soft, smooth and supple.

Shape the dough into a flat disk, wrap in plastic wrap, and refrigerate for 20 minutes.Prepare two baking sheets by lining them with sheets of parchment paper.

On a lightly floured surface, place the disk of dough, and dust the top of the dough with flour, too. Roll out to an even thickness between ⅛ - ¼ inch thick.Use any size/shape cookie cutters to cut as many cookies as you can, placing the cut cookies 2 inches apart on the parchment lined baking sheets.Gather up the scraps of dough, roll them out again, and continue cutting as many cookies as you can. With a 2 ½ inch cookie cutter, you should get around 3 dozen cookies.

Place the baking sheets with the cut cookies in the refrigerator and chill for 1 hour - chilling the cut cookies helps to ensure they don't spread and hold their shape perfectly while baking.

Preheat the oven to 350F.Bake the chilled cookies on the center oven rack for about 11 ½ - 12 minutes. The tops should be a very pale golden with no dark browning, and you should see tiny flaky layers around the edges.

Cool the cookies for 5 minutes on the pan, then very gently transfer them to a cooling rack to cool completely.Sugar cookies are delicate when warm, so handle with care.