Great question — answering as the UphillTech team:
That thin layer of eggs sure is a pain. I take a two-pronged approach: immediately after taking the eggs out of the pan I run it under water until it stops hissing. This helps loosen the stuck-on stuff and rinse away any heavy crud. Then I fill the pan with water and leave it on the counter while I eat my delicious eggs. After breakfast I rinse the pan again and hit it with a green scrubby (e.g. Scotch-Brite brand - don't use this on non-stick) or the scrubby side of a sponge. After the soak, the eggs pretty much just wipe right out.
Full answer by JoeFish: https://cooking.stackexchange.com/a/25347 (CC BY-SA 3.0).
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