1. To prepare cookies, place the first 3 ingredients in a large bowl; beat with a mixer at medium speed until well blended. Add 1 teaspoon vanilla and egg; beat until well blended.
2. Weigh or lightly spoon flours into dry measuring cups, and level with a knife. Combine flours, cocoa, and salt, stirring with a whisk. Add the flour mixture to sugar mixture; beat at low speed just until flour is incorporated.
3. Place half of dough between two sheets of plastic wrap; roll to a 1/4-inch thickness. Repeat procedure with remaining dough. Place dough (still wrapped in plastic wrap) on a baking sheet. Chill 45 minutes or until firm.
4. Preheat oven to 375°.
5. Cut dough with a 2 1/2- to 3-inch cutter to form 40 cookies (reroll scraps as necessary). Arrange 20 cookies on a baking sheet lined with parchment paper (keep remaining cookies chilled). Bake at 375° for 9 minutes or until set. Cool on pan on a wire rack 5 minutes; remove cookies from pan. Cool completely on wire rack. Repeat procedure with remaining dough.
6. To prepare icing, combine 1 tablespoon water and meringue powder in a medium bowl, stirring with a whisk until smooth. Add milk and 1/4 teaspoon vanilla. Add powdered sugar, stirring until smooth. Spoon icing into a small zip-top plastic bag. Snip a tiny hole in one bottom corner of bag; pipe designs on cookies.