So how visible is the side? would a 3/4" or 1" or so circle of wood look really bad? If not, here's what you do:
Go back on the cabinet side to the point where the particle board is still 'good', if you want to call it that. Get a wood boring bit & bore a 3/4" (or larger) hole, which you will then plug with a piece of dowel, glued into place. Plane off the end of the dowel, and paint it, stain it, or hang a sign over it to cover. Now, fill the space vacated by the busted particle board with epoxy, and let that harden good. Get a screw long enough to reach well into the dowel, drill a pilot, and run the screw into the dowel. The screw won't pull out of the dowel, the dowel probably won't pull through the particle board any time soon, and you're ready for company!