Last month I did a job that required cutting out an approximatly 3 inch diameter disk from a 0.100 thick Al sheet. I used what I have seen called a trepanning tool/circle cutter/hole cutter/? Basically a shaft with a central twist drill and a horizontal cross bar that holds a single point 3/16 square tool bit vertically parallel to the central shaft. The horizontal bar slides to adjust the radius of the cut. The tool bit slides vertically as well. I set the bit to contact the work after the central twist drill had gone through the sheet. Ran my Clausing 8520 as slow as it would go and fed down with the quill. Kind of scary with the tool whipping around in a circle and the screaming of the scraping action of the tool on the Al. Used WD40 as cutting lube which helped. Had quite a bit of catching where the tool bit jammed in the work. The belts on the 8520 were a little loose so the tool stalled and the motor spun. I think, a good thing in this instance. I found I had to grind the tool to a narrow chisel profile to reduce the load on the machine and stalling although increasing belt tension might have worked too. Also the narrow tool had an easyer time fitting into the groove being cut. I also reversed the tool bit
180 deg. and ran the spindle in reverse to widen the groove for a little tool relief as well. The 0.1 inch sheet was screwed down to a piece of MDF as was the disk to be cut out. That assembly was clamped to the table.I think that the rough out with an end mill and clean up by boring bar would be a better way to go in 3/8 thick material due to the trouble I had with catches and stalling. Of course a more manly milling machine might have made my method work like a hot knife through butter. My end product was the disk and from the OP's text I assume his is the hole. It worked out and my friend was happy with his big washer. I've seen the circle cutters at decent hardware stores and may have even seen them in the tool dept at Home Depot.
Disclaimer: I'm pretty new at this so take whatever I write with as much skepticism as you feel comfortable with.
Bill