Preheat an oven to 375 degrees F (190 degrees C). Combine flour, cocoa, baking powder, and salt together in a bowl.
Beat the butter, shortening, brown sugar, and white sugar with an electric mixer in a large bowl until smooth. Beat the whole egg into the butter mixture until completely blended, then beat in the vanilla extract and peppermint extract with the egg yolk. Mix the flour mixture into the wet ingredients until just incorporated. Fold in the chocolate chips, mixing just enough to evenly combine. Scoop up the dough by heaping teaspoons and roll into 1/2-inch balls. Place 2 inches apart onto ungreased baking sheets.
Bake in the preheated oven until the edges of the cookies become firm (the center will be slightly soft), about 10 minutes. Cool on the baking sheets for 10 minutes before removing to cool completely on a wire rack.