It's the disconnector!!!!
Really, it is.
It has an over developed sense of responsibility in keeping your rifle from slam firing.
On a working ar-15, this is what holds the hammer away from the trigger until the triger is released. When you release the trigger you get the same click.
To fix the problem, you need to remove the hammer, and then pull the trigger pin slightly out until the disconnector is free. The part you see catching the hammer is a little long.
If you hold it in the same orientation it would be if you were firing your rifle, (upright, like its still in the rifle) then you can stone the hook part thats catching, perpindicular to the ground, a little at a time, reinserting, and checking for clearance.
Or, send it to a good gunsmith,(many recommendations can be found here on the board) and let them install a match grade trigger. You'll love your rifle so much more with a perfect trigger.
HTH
WL
edited for typos