Saw a social media post about this so I tried it today. Instead of putting some butter in the hot frying pan, pour a little heavy cream and then drop the egg in that. Once the water evaporates in the cream, you are left with butterfat and milk solids that start to brown like browned butter. The eggs definitely didn't look pretty but had a really good flavor from the browned milk solids and they had a very soft texture.

Definitely worth trying some time if you like to eat eggs.