I just realised that you all probably don't know what a lamington is. I think it's an Australian thing.
en.m.wikipedia.org
The general thing is to use sponge, but my mum always used butter cake. She didn't fill hers either, but it's common for them to have a thin layer of jam and cream at the midpoint.
The chocolate on the outside isn't buttercream. The process is more like, say, crumbing a piece of meat. The cake goes into a liquid mix of cocoa powder and water, when all sides are covered, it immediately goes into dessicated coconut until it's covered in that, and then it goes on a wire rack to dry. So you have a wet hand and a dry hand, like you do for crumbing a piece of meat.