Egg is pretty normal on things in Australia. Seems in order for something to be Australian cuisine they just slap an egg and some beetroot on.

Non savoury sweet desert pizza is weird to me , basically a bad waffle in my eyes

Egg on a pizza is quite common here.
Usuallly it is called "O sole mio", and has an egg in the middle. It often comes with either sausage or ham, and sometimes with some spinach.

if you're making your own - and you've got time so why aren't you ?

I like to use different types of pesto as a base instead of a tomato sauce.
lidl does three different types for a few pence a jar.

Keeps quite well and just stops all pizzas tasting of tomato puree

I've seen the egg ones, not yet tried it. I tend to make my crust fairly thick and fluffy so cook for about 20 mins - an egg would be hard as  a bullet by the end. So not actually tried it.

If you're going for the 5 minute blast furnace variety, eggyolk should still be runny. That would work really well.

Cheese wise I use low fat mature cheddar.
Has all the stringiness of of mozzarella but plenty of flavour as well. I just don't see the point of mozzarella