In a large bowl, stir together flour, sugar, baking powder and salt. Cut in the butter using two forks (or your hands) until the mixture resembles crumbs.
Set aside 1 Tbsp of the beaten eggs. Add the rest of the eggs, vanilla extract and peppermint extract to the flour mixture and knead until the dough comes together.
Turn the dough out onto a clean surface and divide into four even pieces. Roll each piece into a long log, approximately 8 inches by 2 inches. Place logs on greased cookie sheets and brush the tops with the remaining beaten egg.
Bake the logs for 20 to 25 minutes. Remove from the oven and cool 10 minutes. Keep the oven on!
Once cooled, cut the logs into 1/2” slices and place cut-side down on the baking sheets. Return to the oven and bake for 15 minutes. Cool for 15 to 20 minutes.
In two small bowls, melt the chocolates separately in the microwave in 30 second intervals, stirring between, until smooth. Add each melted chocolate to it’s own Ziplock bag and trim the corner. Drizzle the cookies with both chocolates and sprinkle crushed candy canes on top.
notes
If you live in a drier area, you may have to add 1 Tbsp of water to the dough to bring it together (particularly applicable in Denver!).