Whisk flour, cocoa powder, baking powder, salt and baking soda in medium bowl until blended; reserve.
With mixer on medium-high beat butter, brown sugar and granulated sugar until combined and blended, about 3 minutes.
Add eggs and vanilla and peppermint extracts and beat until fluffy, about 2 minutes.
Add reserved flour mixture and beat until a thick dough forms.
Using wet hands, shape into 1-inch balls and chill 30 minutes.
Meanwhile, preheat oven to 350F.
Arrange cookie dough 3 inches apart on ungreased baking sheets.
Bake 14 minutes or until cookies are set. Cool 3 minutes on baking sheets. Transfer to wire rack and cool completely.
Melt white chocolate in microwave for 45 seconds. Stir and repeat in 10 seconds intervals until melted and smooth.
Dip cookies halfway in melted white chocolate and top with crushed candy cane pieces. Return cookies to wire rack until chocolate has set.