Can you restore the budget that I accidentally deleted in November 2016?
Yes, I can. But, it takes a while to restore since I have to mount the backup data to a special invironment, run your account, 'hack' into your account, and export the budget.
Takes a couple hours. Is there a couple months+ of budget data in it that would make it worthwhile to extract?
Actually, in that vein, do you know off hand what the address for a request to get a budget in export/import format would be? I'm thinking of setting up a cron job with a curl request (presumably with an authentication cookie) to auto-backup my budget .json locally every day (now that my wife and I are sharing one account, things could unintentionally go haywire).
Haha

The exporting process is currently done in the browser, not through a rest endpoint or anything
That being said, it would be easyish to make a script to export your budget data. You'd have to strip the backup logic out of the frontend code and pass it username+password+budget id => .json backup file.