Combine flour, cocoa, baking powder, baking soda, and salt in a small bowl and whisk until combined and any lumps are broken up. In a large bowl, beat butter and sugar together using a mixer on high speed. Add vanilla and egg and beat until combined. Set mixer to low speed and slowly add flour mixture. Beat until just combined and fold in chocolate chips.
Drop tablespoons of batter two inches apart onto a baking sheet coated with cooking spray. Bake for 12 minutes or until set. (Don't overbake or the cookies won't be chewy!) Cook on baking sheets for 5 minutes, then finish cooling on wire racks.
Put broken candy cane pieces on plate or in shallow dish. Spoon approximately 2 tablespoons of ice cream onto one cookie and press another cookie onto the top. Roll edges of sandwich in candy cane pieces. Freeze immediately.
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