It could be one of several things but two come to mind. The first would be a backer for a drawer pull and the second would be a door handle escutcheon.
If were a backer for a drawer pull, would it be considered "high-end" and part of wealthy property owners estate or would it be middle grade to low end?
I found an example from one of those "vintage hardware" sites, of a brass round backplate. Not really vintage just re-productions. I may be looking at it wrong, but I believe the initials are "WJ". Which so far has not been any help... (roughly 2.5")
Here is different backplate that has the "screw hole" attached. Yours seems to have some damage (pointed out) where the screw hole might have been?