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 fluffy, 3 minutes. Beat in the whole egg and vanilla and peppermint extracts. Reduce the speed to low; beat in the flour mixture. Increase the speed to medium high; beat until combined.
Transfer 1/3 cup dough to a medium bowl and 1 cup dough to a separate bowl; leave the remaining dough (about 2 cups) in the mixer. Using a rubber spatula, fold 1 teaspoon white food coloring into the 1/3 cup dough. Fold 3/4 teaspoon red food coloring into the 1 cup dough. Fold 3/4 teaspoon green food coloring into the 2 cups dough. Transfer each color of dough to a separate piece of plastic wrap, flatten into disks and wrap. Refrigerate until firm, 1 to 2 hours.
Line a baking sheet with parchment paper. Unwrap the white dough and roll one-quarter of the dough into a 12-inch rope; shape the remaining white dough into a 1-by-12-inch rectangle. Place both on the prepared baking sheet; freeze until ready to use. Shape the red dough into a 1-by-12-inch rectangle (it will be thicker than the white one). Remove the white dough from the freezer, brush the top of the white rectangle with the beaten egg white and top with the red dough, pressing to adhere. Use your hands to shape the top of the red dough to make it triangular. Attach the white rope to the top of the red dough (it should look like a Santa hat from the side of the dough block). Freeze until firm, 20 minutes.
Meanwhile, roll the green dough between 2 pieces of parchment paper into an 8-by-13-inch rectangle. Remove the red and white dough from the freezer and place on the green dough. Wrap the green dough around the hat, pressing into a triangular log. Roll the log in green sanding sugar. Wrap in plastic wrap and refrigerate until firm, at least 4 hours or overnight.
Position racks in the upper and lower thirds of the oven; preheat to 350˚. Line 2 baking sheets with parchment paper. Unwrap the dough; trim a thin slice off each end. Slice the dough 1/4 inch thick; arrange 2 inches apart on the prepared pans. Bake, switching the pans halfway through, until the cookies are firm but not browned, 15 to 18 minutes. Let cool completely on the pans.