In Bastion, I'm gunning for the top prize at the Breaker's Barracks, where I need to pop all the balloons in the room in as few shots as possible.
What's the best solution to this puzzle? By this I mean the solution which pops all the balloons in as few shots as possible.
Once you've upgraded the Breaker's Bow, you can do it in 5 shots and get the first place reward.
For ease of hitting each row, I recommend charging up a shot, getting into position, then cycling through the targets until you are targeting the last balloon in the row. It makes getting the first 4 rows much easier than trying to eyeball your aim when locked on the front one.
As for theoretical best, you might be able to do it in 4 shots:
The problem with that method, however, is that you can't use the "aim for the last balloon" trick described earlier since they all won't fit on the screen. As such, the difficulty is increased... and in the end, it's not necessary to get first, anyway.
Listen to the music when charging your bow shots. 1, 2, 3, shoot. Put yourself above the balloons, not below. For your first shot, knocking out four balloons in one go (not horizontally, but diagonally) is helpful for making aiming easier for your second-to-last shot. Look at the shadows when aiming, not the balloons.