In a small sauce pan, over medium heat, combine the candy canes, and 1/8 cup of butter. Bring the liquid to a low boil. The candy canes should be melting by now.
Stir in the heavy cream, turn the heat to low, place the lid on the pan, and let simmer, stirring occasionally until the candy canes completed melted. About 5 minutes.
While the candy canes are melting, in the stand mixer, combine the white sugar, brown sugar, and remaining butter. Cream together about two minutes.
Add in the melted candy canes. Mix together on high for one minute, to air-rate the butter and sugar.
Add in one egg, mix together for 30 seconds, then add in the vanilla extract and peppermint extract. Mix together on high for one minute.
In a separate bowl, combine the flour, cocoa powder, salt, and baking soda. Then add the dry ingredients to the mixer.
Starting on low, gently increase the speed of the stand mixer, until the dry ingredients are mixed in with the wet.
Fold in the white chocolate, and semi-sweet chocolate chips.
Preheat the oven to 375° Fahrenheit.
Use a spoon to scoop out the cookie dough, and use your hands to roll the cookie dough into balls about 1 1/4" in diameter. Refrigerate the cookie dough for 20 minutes.
Place 5 cookie dough balls on a cookie sheet, one on each corner, and one in the middle.
Bake for 14 minutes.
Remove from the oven and let cool for 14 minutes.
Transfer to a cooling rack. Serve warm, or save for later!