Step 1

Beat butter with a heavy-duty stand mixer fitted with paddle attachment on medium speed until creamy. Gradually add granulated sugar and brown sugar, beating until light and fluffy, about 3 minutes. Add egg, and beat until blended.

Step 2

Stir together flour, baking powder, and salt in a medium bowl; stir together milk and vanilla in a small bowl. Add flour mixture to butter mixture alternately with milk mixture, beginning and ending with flour mixture. Beat on low speed just until mixture is blended after each addition.

Step 3

Divide dough evenly into 2 portions; shape each portion into a 12-inch-long log. Roll 1 log in 3 tablespoons demerara sugar until evenly and completely coated. Repeat with remaining dough log and demerara sugar. Wrap each log in plastic wrap, and chill at least 4 hours or up to 24 hours.

Step 4

Preheat oven to 350°F. Cut logs into 1/4-inch-thick slices, and place 2 inches apart on parchment paper-lined baking sheets. Bake in preheated oven until bottoms are lightly browned, 12 to 14 minutes. Cool on baking sheets 5 minutes; transfer cookies to wire racks, and cool completely. (Keep unsliced logs chilled while baking sliced cookies.)