Preheat the oven to 350ºF. Line an 8×8-inch baking pan with parchment paper leaving overhang for easy removal; lightly spray with non-stick baking spray. In a bowl set over a pan of simmering water or in the microwave, melt the butter and chocolate; set aside.
In a medium bowl beat the eggs and sugar with a hand-held mixer until light and fluffy, about 3-5 minutes. Add the flour, extracts, espresso powder, and salt. Add the melted chocolate mixture and stir just until blended. Pour the mixture into the prepared pan and sprinkle the top with the chocolate chips. Bake until a toothpick inserted in the center comes out with moist crumbs, about 25 minutes. Transfer to a wire rack to cool completely.
Meanwhile, in a medium bowl, beat together the confectioners’ sugar, butter, milk, and extracts until smooth and creamy using a hand-held mixer. Spread the mixture over the cooled brownies and chill in the refrigerator until set, about 1 hour.
Once the peppermint layer and brownies are cooled, start with the chocolate layer. In a bowl set over a pan of simmering water or in the microwave, melt the butter and chocolate; cool slightly. Pour the chocolate over the peppermint layer and sprinkle with the crushed candy canes. Chill in the refrigerator until set, about 1 hour. Cut into 20 squares. Serve the brownies chilled or at room temperature. Enjoy!