Rickpal,
Since I wanted a roundover on a shallow (3/16) hole, I had to resaw 3/16" off the top of each board, through-drill with a 1 1/8" forstner bit, router the hole's edge with a roundover, then glue it back to the original piece. You can see that on the edge of the middle and bottom shelves. It was a nice demo for the kiddos.
If you don't need a roundover, just drill the board with the forstner bit. It will make a flat-bottomed hole, with a tiny countersink in the middle which can be hidden by the felt (or leather?).