This is a WW2 Plastic Enlisted Cap Badge
The front and back are plastic including the post and the nut.  The post does not have any threads on it.  The nut slides onto it.
From what I read, the eagle is probably on pressed cardboard or foil which is sandwiched in between the plastic.
An example of this can be found in Tonelli's Visor Hats of the U.S. Armed Forces book on page 127.
It measures 1 1/2" in diameter
It is not maker marked
 
Domestic shipping is via USPS Ground Advantage with tracking.