In order for the eyes to end up flat, we are going to do the "wet on wet" technique. This means that as soon as you put a dot of white (or a couple), you put a dot of black right on top, before it dries at all.  The key is to have the right consistency of icing.  I used about a 7 second icing for these. (For clarification on that, see post mentioned above.)  If you wait for the white to dry, you will have more 3D looking eyes.

You can make different style eyes by taking a toothpick and dragging the icing out a little, then put the black dot on.

All of your eyes will need to dry a full 24 hours before they pop right off your parchment paper.  Then you can just stash them in a zip-loc bag until you need them.