Preheat the oven to 350° F. Grease and line an 8-by-8-inch baking pan with parchment, set aside.
For the Brownie Batter: In a medium-sized bowl, whisk together butter, avocado oil, and sugar. Whisk well for 1-2 minutes. Add the eggs, vanilla extract, and peppermint extract. Whisk for an additional 2 minutes, until light in color.
In a small bowl, whisk together the flour, cocoa powder, and salt. Carefully fold the dry ingredients into the egg butter mixture, until just combined. Set aside.
For the Cookie Dough: In a small bowl, whisk the all-purpose flour, baking soda, and salt until combined.
In the bowl of a standing mixer with the paddle attachment, beat the butter and light brown sugar until light and fluffy, 3-4 minutes. Add the egg and vanilla extract, mix until incorporated. Add the flour mixture, mixing only until incorporated. Fold in the chocolate chips and the ¼ cup of candy cane pieces.