Apparently, the majority of the small oval-shaped brass discs that we dig are part of a cufflink.
My guess about Kaleb73816's find:
Most of the time, a cufflink's "connector" was soldered or brazed onto the oval-shaped disc's back. Perhaps in this case the connector was riveted onto the disc, by using the small beveled hole in its center. Or, perhaps the beveled hole held a rhinestone (which makes sense due to the engraved "starburst rays" going outward from the hole), and the cufflink's connector was indeed soldered onto its back. I can't be sure from looking at the photo whether there is a solder-spot on the back of the oval-shaped disc's back, or not.