Whisk the flour, baking powder and salt in a medium bowl. Beat the butter and granulated sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Add the egg and vanilla and peppermint extracts and beat until combined. Reduce the mixer speed to low and beat in the flour mixture, then increase the speed to medium high and beat until well combined.
Divide the dough into 3 pieces. Return 1 piece to the bowl; beat in the red food coloring with the mixer until the color is fully incorporated. Beat the green food coloring into another piece of dough. Leave the last piece of dough plain. Place each piece of dough on a separate sheet of plastic wrap and pat into a 4-inch square. Wrap and refrigerate until firm, at least 1 hour.
Lightly flour 1 piece of dough; roll out between 2 sheets of parchment paper into a 9-by-12-inch rectangle. Repeat with the remaining dough. Refrigerate the dough rectangles until firm, about 30 minutes.
Carefully remove the top piece of parchment from each piece of dough. Flip the plain dough on top of the red dough; remove the parchment. Flip the green dough on top of the plain dough; remove the parchment. Cut the stack in half crosswise. Roll up each half lengthwise into a tight log, using the parchment to help you. Transfer to a baking sheet, cover loosely with plastic wrap and refrigerate until firm, at least 1 hour.
Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Trim the ends of each log; slice into 1/4-inch-thick rounds and divide between the prepared baking sheets. Sprinkle with coarse sugar. (If the dough is too soft to slice, refrigerate until firm, 15 to 30 minutes.) Bake, switching the pans halfway through, until the cookies are just firm, 12 to 15 minutes. Let cool 10 minutes on the baking sheets, then transfer to racks to cool completely.