When glazing, use a shallow bowl that will accommodate the entire cookie.
Add egg and vanilla; beat 30 seconds.
Add flour, beating at low speed until combined.
Credit:Linda Pugliese, Hector Sanchez; Styling: Heather Chadduck Hillegas, Buffy Hargett Miller
Place dough on lightly floured parchment paper, and roll to 1/4-inch thickness.
Transfer rolled dough and parchment paper to a baking sheet, and chill 15 minutes.
Preheat oven to 375.
Cut dough with lightly floured 2-inch cutters; place 1/2 inch apart on 2 parchment paper-lined baking sheets.
Reroll scraps, and repeat process.
Bake, in batches, at 375 for 8 to 10 minutes or until golden brown around edges.
Cool completely on a wire rack (about 30 minutes).
Whisk together 2 1/2 Tbsp.
water and remaining 2 cups powdered sugar.
Dip top of each cookie in glaze, and transfer to a wire rack in a jelly-roll pan.
Let stand 1 hour or until set.
