How to thank a Director in another org for something sensitive?

It is my opinion that someone at this level helping a more junior employee through a sensitive HR situation is sort of part of the job. Yes, it was nice they did this for you, who is in another org, but their impact should be very broad.

This feels more like you should write and send a hand-written note thanking them personally. From there, sharing with their manager or HR in their space seems too convoluted. At most you could say “I had a very difficult and sensitive situation I cannot discuss explicitly, and person X went above and beyond to aid me in navigating this. Without help on this matter <fill in, would you not work there?>.” That seems to cover like… a great thing was done, but it cannot be discussed.

Hey!

Emails sound appropriate in this situation.

To the director's manger:
"Hello, I am Name from Department, working under/with ManagerName. I wanted to write this email to recognize how DirectorName went above and beyond in personally assisting me with a sensitive situation. He was helpful, empathetic, and connected me with the right people. His support is greatly appreciated and a positive reflection of our company's values. "

Keep it short and simple. I doubt he'll ask for details. If he does, say you appreciate the follow-up but mention it was a sensitive HR situation and reiterate that you wanted to highlight your gratitude.

To the director:

"Thank you again for your help with my situation. I am grateful to have had your counsel and support during that time. I have forwarded my appreciation to DirectorManagerName. Let me know how I can ever assist you."

You can go into detail more to express your appreciation but it's not necessary. If there's some formal way of giving kudos to someone during evaluation, I would offer to fill that out.

I wouldn't overthink it.

Good luck!