Unwrap mints and place in a food processor; blend until mints are reduced to dust and small pieces.
Melt 2 cups white chocolate in a microwave-safe glass or ceramic bowl in 30-second intervals, stirring after each melting, 3 to 4 minutes. Add crushed peppermints and fold in until well mixed. Spread mixture evenly onto prepared baking sheet.
Melt dark chocolate chips in the previously used microwave-safe bowl in 30-second intervals, stirring after each melted, 1 to 2 minutes. Spoon dollops of the melted dark chocolate onto the white chocolate mixture and spread evenly. Place remaining white chocolate chips on top at desired intervals. Sprinkle bark with red sugar.
Refrigerate bark until completely solid, about 4 hours. Peel off waxed paper and break bark into pieces.