Mix together the cornstarch and powdered sugar in a medium bowl. Line a 9x5 or larger baking dish with parchment paper and then dust with a layer of the cornstarch/powdered sugar mixture, set aside.
In a small bowl, add the 1/2 cup of cold water and sprinkle the gelatin over it. Allow the gelatin to soften for about 10 minutes.
Meanwhile, add the other 1/2 cup of water and the granulated sugar to a medium saucepan over medium heat. Stir and cook until sugar is dissolved.
Add the gelatin and stir until fully mixed into the sugar water. Bring just to a boil and then remove from heat.
Pour into the bowl of a stand mixer and allow to cool for a few minutes.
Add the vanilla and peppermint extract, then beat on medium - medium high in the mixer for 10-15 minutes, until mixture is white and glossy and has doubled in volume. (You may need to slowly raise the speed to avoid splashing at first.)
Pour into prepared pan (the smaller your pan is, the thicker the marshmallows will be). Use a spatula to smooth the top. Dust with a thin layer of the cornstarch/sugar mixture. Allow to set for several hours, until the marshmallows are not sticky to the touch.
Remove from dish and use a slightly greased knife or pizza cutter to cut into squares. Dust each square in the cornstarch/sugar mixture, then dust off excess powder.
Melt the chocolate over a double broiler, stirring until melted.
To dip each marshmallow, dip half of the marshmallow into the chocolate, then lift it out and allow extra chocolate to run off.
Place each marshmallow on a sheet of parchment paper and allow chocolate to start to set for 5-10 minutes* (but not all the way).
After chocolate is partially set, stir the crushed candy canes to make sure it isn't sticking together (the crushed candy canes harden quickly). Then dip the chocolate side of the marshmallow into the crushed candy cane to coat the chocolate. Repeat with each marshmallow ( you may need to stop a few times and stir the candy cane to make sure it isn't sticking together)
Allow the marshmallows to sit for until chocolate is fully set.