Whisk cassava flour, cocoa powder, baking powder, and salt together in a bowl.
Combine butter and coconut sugar in a large bowl; beat with an electric mixer until fluffy. Add coconut milk, maple syrup, egg, vanilla extract, and 1/2 teaspoon peppermint extract to the butter mixture; beat until well combined. Add flour mixture to the butter mixture; stir until batter is evenly mixed.
Heat donut maker according to manufacturer's instructions. Pipe batter into donut forms using a pastry bag or a clean resealable plastic bag with one corner cut off.
Bake according to manufacturer's instructions, about 4 minutes. Transfer donuts to a rack; cool completely, about 15 minutes.
Melt chocolate chips, coconut oil, and 1/4 teaspoon peppermint extract in a double boiler, stirring until glaze is smooth. Spoon into a shallow bowl. Dip top of each cooled donut into glaze and then into crushed candy canes; return to cooling rack. Repeat with remaining donuts.