To be fair, starting a second essay with no feedback is a losing situation for the students. Perhaps the second should have been assigned after returning the first, and then pick up the pace?

His e-mail is not really demanding after all. Except for the word "promptly", which is indeed inappropriate in this context, "Please let me know" is a reasonably polite request. If the student is not a native speaker, he or she might not feel the difference between "promptly" and "whenever it is possible for you". You can't expect that they know the email etiquette right from the start. Where would they have the opportunity to learn it The student might have missed your announcement about 2 weeks, which also happens often.

He might be assuming people get feedback at different times and is trying to get an advantage by being pushy and getting his own feedback before starting the second essay while others will have to make substantial rewrites

@Vosoni I disagree, the email comes off as demanding and rude. While I agree the use of "promptly" sounds like a non-native speaker, it's still rudely written.

Just a friendly reminder, especially for future visitors, that such a situation frequently arises with exchange students, who do not yet speak the language fluently. In certain cultures direct, informal speech is the normal.

If you decide to address the issue, I would advise you to be clear about what offended you in the email (presumably the word "promptly"), rather than make a vague point about "issuing demands to academic staff regarding how we conduct our teaching work". If the student didn't understand that it was rude of them to ask you to "promptly" do something, they are unlikely to correctly identify the offending part of their email based on this. You could e.g. write: "Please note that it is somewhat inappropriate to request "prompt" feedback when I have explained in class that" etc.

@Matsmath Nevertheless, exchange/foreign students are expected to follow norms in their host country (note that OP hasn't mentioned their country), so your answer could include a bit on how to address this with a foreigner vs. lifelong resident.

@ Azor Ahai - I am just saying that the student may not realize it and probably just lacks understanding of email etiquette.

A lot would depend for me on whether this student is a native English speaker. If they are then yes, I would address their tone. If not, I would probably give them benefit of the doubt. If there is a pattern of this student, or students in general, not respecting you because of your gender, then that might justify being a bit more stern as you have above.

It's an overreaction coupled with speculation about the student's motives. In the best interpretation the student is motivated about the course and wants to do a good job with the second essay by using the feedback from the first, but they are not very good at communicating it. In the "worst" interpretation the student does not respect your authority. I do not believe that a passive-agressive e-mail is beneficial in either case. Personally when I teach students I am unconcerned about their opinions of my authority, my job is to assist them in learning. In fact they should question my authority