I like the idea! It's a good implementation of recursion.
The only real downside to the design is that you can't know in advance what will come in the future, so initially you just have to go through the full 3-mouse cycle to then start figuring out the puzzle. It would be better if you saw the spawn positions of the mice, and their initial walking direction right from the start, as well as the cheese positions. This would let you plan in advance.
Other minor things are: Sometimes the mouse is a bit too sensitive and a small misplacement of a block can result in mice getting stuck. An improvement here would be to snap the blocks to a grid when you let go of the mouse or something.
But all in all, it is really cool to see you replaying previous levels solutions as part of the solution for future levels. I think you really have a good idea going there that just needs a bit more fleshing out!
Good job!