We've discussed boxes, sub-doors and all manner of hard-sided solutions, but the shape of the cupboard makes all those awkward. The best solution came down to a bag of some sort, so the other day I decided to just go for it.
I made the bag out of some light canvas that has been sitting around for years, and bound the top edge with some green bias tape.
I made a cutout in one side of the top edge as an opening for the recycling to go in, then added four webbing loops at the corners.
I bought four heavy-duty cup hooks from Canadian Tire and screwed them into the corners of the cupboard. And viola! No more avalanches!