Pre-heat the oven to 325 F. Line several baking sheets with parchment paper. Set aside.
Add almond meal and 1/3 confectioners sugar to a food processor fixed with a metal blade. Run until a fine powder forms.
Combine the ingredients from the food processor with the remaining confectioners sugar. Sift four times, discarding any tiny lumps or almond meal “pebbles” between sifting. Set aside.
Using a stand mixer fixed with the whisk attachment, whip the room temperature egg whites on medium speed until foamy. Add the granulated sugar one tablespoon at a time.
Once the mixture thickens, stop the mixer. Add cream of tartar, scrape down the sides of the bowl, then turn the mixer back on, increasing speed to medium high.
Whip the mixture until a peak at the end of the whisk attachment holds its form. (You can test this by turning off the mixer and checking to see if the mixure at the tip of the whisk attachment doesn’t drip.)
Affix a pastry bag with a 1/2 inch plain round tip. Fill the bag 3/4 with batter. Pipe into approximately 1 inch rounds on the lined baking sheets. You want to pipe so that the bag is perpendicular to the baking sheet, not at an angle. At the end, flick your wrist quickly to the side so a point doesn’t form at the top of the macaron. (If a tip does form, no worries, you can smooth it out with a small spatula).
Gently tap the baking sheet on your work surface to get rid of any air bubbles potentially stuck in the batter. Let the prepared baking sheets sit at room temperature until a dry shell forms on the macarons – test by touching the batter with your fingertip. If your fingertip doesn’t stick, they are ready for the oven. The “resting” phase can take 20 minutes to 40 minutes, depending on the humidity in the room.
Bake the macarons, one sheet at a time, in the lower third of the oven for 12-13 minutes, turning halfway through. When done, the tops will be firm and the middle/feet won't jiggle when you gently touch the top.
Remove macarons from the oven and let cool for 5 minutes. Gently peel the macarons off the parchment paper and transfer to a wire rack. Allow the macarons to cool completely before filling.
To prepare the peppermint buttercream, bring 2 cups of water to a boil in a saucepan. While that is heating, whisk the egg whites and sugar in a heatproof bowl. Set the bowl over the saucepan, ensuring that the bottom of the bowl isn't touching the water.
Whisk the mixture occasionally as it heats, for about 4-6 minutes until the mixture thins and is hot to the touch.
Pour the heated mixture into the bowl of a stand mixer, fitted with the whisk attachment. Whip on medium for 4-6 minutes until cool, white, and foamy.
Reduce the speed to low and add the butter, one tablespoon at a time. Increase the speed back to medium and beat for another 4-5 minutes until the buttercream thickens and is smooth. Remove bowl from the stand mixer and stir in the peppermint extract.
Fill another piping bag affixed with 1/2 inch plain round tip with the buttercream. Arrange the macarons by two's, top side down, making sure to match sizes best you can. Depending on the size of the macaron, squeeze a nickel to quarter-size amount of buttercream onto the bottom of the macaron. Lay the second macaron on top, and very gently, press together so that the filling expands but doesn’t come out the sides. Repeat for all remaining macarons.
In a medium-sized bowl, microwave the white chocolate chips on 15 second intervals, mixing in between until melted and smooth.
Fill a ziploc bag with mini candy canes (or, if they are individually wrapped, keep them in the wrapping.) With a mallet or rolling pin, gently pound the candy canes until they are crushed into little pieces - but not powder. Pour the pieces onto a small plate.
With a knife or small spatula, very gently cover the top of a macaron with a thin layer of white chocolate.
Gently dip the macaron white chocolate side down into the candy cane pieces so the top is covered. Place back onto the baking sheet to set. Repeat for all macarons.