The ice maker 'ice mold' tray had it's slick surface coating peeling away. There was a lot of peeling - it was bad. Although I couldn't see in to verify what was truly happening, the thought was that the peeling was somehow diverting the water out of the tray, causing it to leak and form an icicle into the ice bin.
Even if this wasn't the fix for the icicle, the ice mold still needed replacing because of the coating contaminants in the ice cubes.
I disassembled the ice maker, replaced the ice mold, and now it works perfectly. No icicle.