How can I get the backend user's name and output it on the backend form field using a partial.
Every partials which are render in back-end has "user" object within in it.
and this object is pointing to current logged in back-end user.
so if you want to add that user's fisrt_name in partial you can do something like this.
<?= $this->user->first_name ?>
but from you code I can sense that you want to use use-name and try to use it as key for $value array and get inner value of $value array
so first {{ }} this will not work inside php tags, instead good news you can directly use php code there :)
your solution:
<?= $value[$this->user->fisrt_name] ?>
you can replace attribute first_name to another column attribute which will appropriate to you.
if any issue please do comment.