No, it would not be possible, the etching process would destroy extant features on any silicon. There are multiple steps to it and some will destroy the work of the steps that come after, so putting a finished chip through would remove the existing features.

Generally fabrication is done on a circular 300mm wafer, printing a grid of identical chips on it that are then cut out. It would be highly inefficient to do each chip one at a time, doing one small chip or 50 chips at once on one wafer takes the same amount of time and effort.

And the bare silicon left after the first few steps isn’t really worth much. You can go buy a blank high end 300mm wafer for like 81 bucks online. The material value of the silicon is nothing compare to the machine time.