Sift the cake flour, almond flour, cocoa powder, baking powder, and salt into a bowl and whisk to combine.
In a large bowl, beat eggs, sugar, and peppermint oil with a hand mixer on a medium speed until it has at least doubled in size, about 3 minutes. Add the melted butter and stir to combine.
Add the dry ingredients and gently fold until just about combined. Add the mini chocolate chips and fold gently to combine. Cover the bowl and chill until cold, about 1- 3 hours.
About 15 minutes before you are ready to take the dough out of the fridge, preheat the oven to 350°F and line a baking sheet with a silicone mat.
Turn the biscotti dough out onto the prepared baking sheet and shape into a 10- 12 inch log. The dough will be sticky. It helps to wear food safe gloves, use cake scrapers, or lightly oil your hands.
Bake the log for 35 - 40 minutes, or until light golden brown. The biscotti is completely baked when the top does not dent when poked and a skewer comes out clean (though you may see melted chocolate). Cool on the baking sheet for 15 minutes before carefully transferring to a cooling rack. Cool for about an hour or so.
When the log is cool, preheat the oven to 325°F. Slice into 1 inch cookies on a slight diagonal. Place the cookies upright on the lined baking sheet. Insulate the bottom of the baking sheet by placing it inside a second baking sheet of the same size. Bake for 40 minutes, rotating the pan halfway through. Place cookies on a cooling rack and cool completely.
Melt the semisweet chocolate gently by using a double boiler or the microwave (50% power, stirring every 20 seconds or so). Once the chocolate is mostly melted, continue to stir away from the heat until smooth. Place a large plate in the microwave and heat for 30 seconds.
Pour the chocolate onto the warmed plate in a rough rectangle shape. Dip the bottoms of the biscotti into the chocolate one at a time. Wiggle slightly to push a bit of chocolate up the sides. Use a small spatula to remove any excess chocolate from the bottom of the cookie.
Over a separate bowl or plate, sprinkle crushed candy canes onto the melted chocolate (sides only). Set on wax paper until the chocolate is firm.
To store, place in an air tight container on the counter for up to 2 weeks. Serve with your favorite hot beverage.