Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
In a bowl, whisk together flour, baking powder, and salt. Set aside.
In another bowl, beat butter and sugar until light and fluffy. Add egg and vanilla extract, and beat until combined.
Gradually add the flour mixture to the butter mixture, mixing until just combined.
Divide dough in half. Add peppermint extract and green food coloring to one half. Add cocoa powder and milk to the other half. Mix each until combined.
Roll out each dough half into a rectangle about ¼ inch thick. Cut each rectangle into strips.
Arrange strips in a checkerboard pattern, pressing gently to adhere. Cut into squares.
Place cookies on prepared baking sheets. Bake for 10-12 minutes, until edges are set. Cool on wire racks.