I found this thread from two years ago with a response from the Yeson Center:
https://www.susans.org/forums/index.php?topic=149644.0"Dear Paige
When to have trachea shave doesn't matter. It anyway affects your voice if they make a high resection on your thyroid notch.
That's why we do not recommend it. If you really want to do that, we recommend you should receive the surgery with us to minimize the damage to your vocal folds."
And this:
"Dear Paige
Thank you for resending your e-mail :-)
Regarding your inquiry, Dr. Kim ,as a professional surgeon, will perform the thyroid cartilage shaving on you.
To not affect the outcome of VFS surgery, only a little ressection on your thyroid notch will be made.
Sorry before and after photos are not available. We will be glad to answer your further inquiries
Thank you and warm regards,"