I am assuming that producing these documents is something that involves a bit of work. My answer would be different if it is timesheets or similar document.
Shaming people and publicly pointing out their mistakes is a great way to kill morale. Huge faux pas on behalf of the sender. You can be sure that most people who got that email are as uneasy as you about it.
If I got such a request/demand with a deadline I would go to my closest manager because she is the one who decides on my priorities. I would not even consider working on the task without clearing it with her first. So my back would be clear when the second email comes. And when my back is clear, who looks like the fool?
This would be my reply:
I discussed this with Boss and we will not be able to deliver these documents by the deadline. Feel free to talk to her if you need her to change my priorities.
This email would point out that I have other important work tasks, follow a planned process, and also give a course of action if the request is indeed important.
I have sent similar messages many times in my career because getting late requests out of process seems to be the favorite pastime for some people. I understand that others also have a job to do, and I am happy to sidestep the process for quick tasks when I have the time. I love to make someone else's day a little easier. But if the task is demanding or I have other deadlines to meet I would not hesitate to involve my own manager.