Dragoon or Civil War Cavalry Bridle Boss .  Notably, this example has a neatly scratched "H" added to it by the soldier.  Was this his company?  Did his horse's name start with an "H"?  or was he simply ready to ride it on "H"-O-M-E?!
Bad humor aside, this still has the original leather on the reverse.  It is 43mm or 1 and 3/4 inches in diameter (which is larger than the size of a silver dollar).