Paper-plus-electronic voting machine
Proposal:
Voting machine has the usual touch screen.
Plus, a transparent window that lets you view a till-roll-style strip. The machine has several 100's of feet of paper, so it need not be opened until the poll is over.
You vote.
Vote is printed in an OCR-able form on the paper.
You confirm the vote (through the window) on the paper is as you voted.
The votes get stored both electronically, and as 4 inches of paper on the roll in the machine.
When a recount is requested, you can take the long reels of paper and count them either in another machine, or, in extremis, by hand.
So, the primary count is electronic, but it can be backed-up by two levels of physical re-count.