I'm going to experiment with making some lampshades, too; some of the species are so beautiful when they glow with the light, so warm.
Veneer is a perfect winter basketry material because it's available any time, and requires so little processing: just pour boiling water over it and let it soak for 20 minutes or so.
When I was first experimenting, I would cut the sheets into strips with my paper scissors. I found that got a little tedious and time consuming when cutting strips for the number of these types of basket that I'm making these days, so I've turned to using a paper cutter. It is very, very satisfying.