Preheat oven to 350 degrees F.
Grease 2 donut pans and set aside.
In a large bowl, whisk together 2 tablespoons of melted butter, granulated sugar, light brown sugar, milk, vanilla extract, peppermint extract, salt, and the egg.
Add the last 3 tablespoons of butter and dark chocolate chips to a microwave safe bowl. Microwave in 10-20 second increments until the butter and dark chocolate melt completely. It should take about 45 seconds. Once melted, mix until you get a smooth texture.
Pour chocolate into the bowl with the wet ingredients. Mix until well combined.
Sift sweet rice flour and baking powder into the bowl. Use a spatula to combine until you get a wet dough.
Spoon the dough into a pastry bag or Ziploc bag. Pipe the dough into the donut pans. I used a donut pan with donut molds that are 3 inches in diameter.
Bake for 9-11 minutes until a toothpick comes out clean.
Let donuts cool in the donut pan for 10 minutes. Transfer to a cooling rack to cool completely.
Once the donuts are cool, make your ganache. Add chocolate chips to a bowl. Heat heavy cream until bubbles start to form around the edges. The cream should be hot not boiling. Pour cream over the chocolate chips. Let the mixture sit for a few minutes to soften the chocolate. Then mix until a smooth ganache forms.
Dip the donuts into the ganache. Top with crushed candy canes and serve.