Let's Play Puzzle Swap!
Or...or not...? ;-;
The music sample at the end was created by my great friend Chase Pugh! It's called "A Walk in the Park", you can listen to it here: https://soundcloud.com/commanderspleen123/walk-in-the-park
To perform this glitch yourself:
1) Press the home button as soon as the bird's chip finishes (immediately before it comes on screen). This will cause it to fly up off-screen, rather than to your mii
2) Press it again as soon as the bird drops the piece. This is hard to time because it happens off-screen and has no audio cue!
The timing is pretty tight, I honestly have no idea how I accidentally bumped into this one. Confirmed working as of Mii Plaza v3.3 (visible in Mii Plaza settings), on the New 3DS. Let me know if you have luck on other models!