Preheat oven to 325° and butter 2 or 3 baking sheets.
In a large bowl with a mixer on medium speed, beat butter, sugar, and vanilla until smooth. In another bowl, whisk together flour, ginger, baking powder, and salt. Reduce speed to low and beat or stir in flour mixture until blended. Beat on medium speed until dough forms a ball. Divide dough in half.
Transfer dough to a lightly floured surface. With a lightly floured rolling pin, roll each half into a 1/8-in. thick round. Cut out cookies with a floured 3- to 4-in. leaf-shaped cutter. Place cookies 1/2 in. apart on prepared pans. Gather excess dough into a ball to reroll and cut out remaining cookies.
With the back of knife, lightly press a decorative vein pattern into cookies. In a small bowl, beat together egg and milk. Brush over cookies and sprinkle with sugar.
Bake cookies until edges are just beginning to brown, 12 to 15 minutes, switching pan positions halfway through baking (if using a third pan, bake it separately). Cool completely on baking sheets, then transfer to an airtight container.
Note: Nutritional analysis is per cookie.