Preheat oven to 350°F with racks in the upper and lower thirds. Line two baking sheets with parchment paper. Whisk together flour, cocoa, baking powder, and salt in a bowl.
Beat butter and 1 ½ cups confectioners' sugar on medium speed with an electric mixer until smooth, 1 to 2 minutes. Reduce mixer speed to low and gradually add flour mixture, beating just until incorporated, 1 to 2 minutes. Add food coloring, vanilla, and vinegar; increase mixer speed to medium, and beat just until dough is crumbly and holds together when squeezed, 15 to 20 seconds.
Squeeze dough into balls (about 2 tablespoons each). Place balls, 1 inch apart, on prepared baking sheets; freeze 10 minutes. Bake, rotating sheets halfway through, until dry around the edges, 15 to 18 minutes. Cool on baking sheets on wire racks 5 minutes.
