Just complete every ACT mission on HARD mode. Unlocked after beating the game normally and accessed by looking under the mission select tab.
You don't have to replay bosses. Hard mode just changes up the enemies, adds more traps, moves switches around and sometimes takes away springs.
NOTE: In the case of "Dusty Desert" the final puzzle is changed by removing some boxes and adding spike balls in places. In some places this actually makes it easier to "S".