Not a lawyer here. Why does an Article III judge need to get involved, and not in other administrative matters, like say, in an administrative immigration case?

edit: further investigation has indicated that it's immigration law which deviates from usual practice of administrative law, which would potentially expect an automatic stay in a context like this.

To the contrary, the order already exists (“stop it”) and this article describes the second part of the process you linked to:

  > The decision of the Administrative Law Judge may be reviewed by the five member board