A "cone" being a Australian/New Zealand slang word for bowl (of your bong) a "breakkie cone" is a phrase describing a toke or two before breakfast. Also known as a Pre-breakkie cone. The post breakkie cone obviously wraps up your breakfast session.
Stoner #1--"Man I am so starving after that breakkie cone I might eat your mom."
Stoner #2--Fuck off bag stand, lets grub.