The outer brown part. What's a good/easy-to-understand to refer to it?
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Some common phrases are crispy edges and burnt edges, and they are easy to understand. These or similar phrases are used in "how to fry an egg" tutorials and egg recipes also.
An egg, sunny-side up, cooked in butter or enough oil with no chewy whites or burnt edges.
It's hard to beat the satisfaction of a perfectly fried egg with its delightfully crispy edges and bright, creamy yolk.
Personally, I've always called it frizz - based on the verb defined there as...
to form into small, crisp curls or little tufts.
But I'm not very specific in my usage. That same word also does me for the little dribs of batter that fall off fried fish (when I'm not collectively calling them frazzles - frayed ends or remnants).