Prepare cookies: Position an oven rack in center of oven and a second rack in lower third. Heat oven to 350 degrees F.
Combine flour, baking powder, baking soda, and salt on waxed paper. Beat together butter and sugars in a large bowl with a mixer on medium speed until creamy, about 2 minutes. Beat in buttermilk, egg, and vanilla until well blended. On low speed, beat in flour mixture in 3 additions.
Drop batter by level 1/4 cup onto 2 large ungreased baking sheets, spacing about 3 inches apart. Using a metal spatula, spread into 2-inch rounds.
Bake for 8 minutes; switch positions of baking sheets. Bake another 7 to 9 minutes or until edges are lightly browned. Cool baking sheets on a wire rack 1 minute. Transfer cookies to rack to cool completely.
Meanwhile, prepare icings: For black (chocolate) icing, melt chocolate chips and shortening in top of a double boiler over barely simmering, not boiling, water, stirring until smooth. Let cool.
For white icing, beat milk, confectioners' sugar, and salt in a medium-size bowl until a spreading consistency, adding more milk if necessary; cover with plastic wrap.
Turn cooled cookies flat side up; brush off any crumbs. Spread 1 tablespoon white icing on half of each cookie. Spread 1 tablespoon chocolate icing on the other half of each cookie. Let dry. Store between sheets of waxed paper in a sealed container.