The flat, round part of your item with the square hole is the back plate or escutcheon for a drawer handle or pull. The other piece is called a mounting post I believe, and is screwed into the wood drawer on the back side of the plate. The hole in the front would accept the end of a drawer handle, probably of the bail type, and there would be 2 sets of this hardware per handle (one for each end). Yours looks similar in basic style to the ones I find at colonial sites up here, but does appear to have a slightly newer look to it so 19thC seems reasonable.