Once you've finished the entire game (after receiving the achievement "Closure") go into the main menu by hitting start and then select "Speed Runs." From here select "Full Game." Now you've got 45 minutes to beat the game (collecting every puzzle piece mind you) this isn't impossible to do but it may take two or three tries at the minimum.
My times when I finished the game in 44:56 (phew!) were...
4:06 to complete world 2's puzzle
8:52 to complete world 3's puzzle
18:34 to complete world 4's puzzle
30:42 to complete world 5's puzzle
39:58 to complete world 6's puzzle
44:56 to complete world 1
The main thing to remember is simply to go as fast as possible and know exactly how to do each level as quickly as possible. I made what I would assume is a perfect run and only just barely finished. Solve the puzzles as soon as you get the last piece and end up in the house, they don't count towards your time taken.
You can't use the menu to do "Return to Clouds" so when you have to go back to level 2-2 to get the two missing puzzle pieces you have to continue through 2-3 and 2-4. This also gets annoying when you screw up on a mission that you can't fix (like Irreversible.)