I'd theoretically put on some sort of head covering to protect my hair, crack the egg into the frying pan, then tape the frying pan to the top of my head and step outside in this godforsaken heat.

theoretically.

you'd need decent balance and willpower to pull this off but it's doable.

again, theoretically.

bald cap that's reflective probably; wear sunscreen

Put the egg on a plate or some other type of pan, balance it on my head, take a magnifying glass and use it to concentrate direct sunlight on the egg until the heat cooks it.

Theoretically.

quick follow up: does said egg need to be fried directly on head or can there be some sort of buffer like a plate or such?

Theoretically, I would put on a crap ton of heat protection for my hair before doing anything else. I would then take a very small plate/tray (something that can withstand immense heat) and tie it to the top of my head. Then, I would fry the egg with a hair dryer. Only in theory, of course, because this is purely theoretical.

I also wouldn't actually try this because I don't know how safe it is. Thankfully, this is all in theory and nobody will actually be doing this, least of all yourself. I'm so glad this is theoretical.

Best of luck to this theoretical person.

Go bald, foil cap, butter, egg, direct sunlight.
I might be burned at the end off the day bur atleast I have my sunny side up

theoretically, if we got flexible with the meaning of "on", then if there were some way you could sit beneath a stove, eg. put a camping stove on a table then sit beneath it, you can theoretically have the egg in a pan on a stove on a table on your head - therefore, the egg is, theoretically, on your head, and being cooked. in theory.