In a large bowl, beat butter with an electric mixer until creamy. Add sugar; beat until combined. Beat in egg, peppermint extract and vanilla.
In a small bowl, stir together flour, cocoa powder, baking soda, baking powder and salt.
Gradually beat flour mixture into butter mixture, beating well after each addition. Cover dough and refrigerate for 45 minutes.
Preheat oven to 375°F.
On a floured surface, roll out dough to ¼-inch thickness. Use a 3-inch-square cutter to cut cookie shapes. Place 1 inch apart on ungreased cookie sheets. Bake for 5 to 7 minutes or until light brown on bottom. Cool.