The Mezzanine

I had quite the time of it when we were in Australia, because I'm allergic to beetroot and it was EVERYWHERE. (As a side note, when onetwomany visited the US for WriterCon and went on a roadtrip with me up to Seattle afterward, she commented that avocado was America's beetroot... we put it on everything!)

I'd never thought of putting egg on a hamburger until I ran across it at Aussie Eats (an Australian restaurant that used to be in Idaho Falls, Idaho... of all places). It had pretty much everything you listed there and was called The Genuine Australian Burger. I never had one (see beetroot allergy above), I'm afraid... I much preferred their meat pies.