I don't care for scrubbing. Too much work, and you end up with a scrubber that looks vile. Try this instead: Rinse the eggy pan, add Washing Soda and 2 drops of dish wash soap. Now push the solids around with a soft spatula. How nicely the residue sort of rolls up and liquifies! Scrape the pan bottom with a steel spatula if necessary. Let it stand for a few minutes. The gunk will come off easily when you rub with a soapy wash cloth under a trickle of running water.
Washing Soda is basically sodium carbonate, is my impression.(I use "Arm & Hammer Super Washing Soda". In my part of the world it comes in a yellow cardboard box.) Maybe Baking Soda would work too? (That is bicarbonate of soda.) I'm going to try the Ammonia idea someone contributed.
Do NOT use non-stick pans! You very well know that the non-stick coating wears off. Bits of that coating end up inside of anyone who eats your cooking. Assuredly, we were not built to ingest those man-made scrapings. For the love of your descendants, don't implant toxic waste in your kids' bodies.