distort maps have their limit aswell, it would turn into a hopeless garble after only a few cuts and getting a texture to look right would be very difficult on pieces, not to mention distort maps dont have collision detection which adds even more complexity. if you use canvases and your objects arent to big (128x128 lets say being the max) the vram usage wouldnt be that big even with many since the more, the smaller the canvases are, and you could probably have up to 20 on screen with a decent card and absolutely no slowdown.
wtv way you do it, it requires some trickery.
something like that game would never use more than 4 "canvases", hence the reason its lag free.
or for example chisel http://www.nitrome.com/games/chisel/
even though it may seem like the world is always being destroyed and eventually it would use way too many canvases, you have to think about it in a smarter way, only 2 or 3 are needed, one or two to recieve "cut pieces", then a giant one to capture it all once it stops moving, and then those 3 keep repeating the cycle, notice how you cant make two cuts at once? its because it needs time to make sure you dont reveal the "cycle" as the pieces converge.