Student essay feedback for a large class
Here are some options/ideas to consider:


Giving just a grade and asking to come to office hours is certainly defensible. I've read that most students don't read extensive feedback.
Commit to giving a grade and very short feedback.
Create a written rubric with check-off categories for various scoring criteria. Require students to staple this as a cover sheet on their submissions. Grade by checking appropriate boxes.
If you have an online learning management system, it may support rubric grading as above, so you can do the check-off grading without additional paper. (e.g., I do this on Blackboard for computer programming assignments).
Set a budget as far as time you can legitimately devote to this task. Keep written records (e.g., spreadsheet) of how much actual time is spent on this overall task. Have the TA track time as well. If time spent is over budget, iterate and cut/modify/streamline something.