Figured out a very streamlined way to get golden goose eggs!
First, tame yourself a female goose friend. You do this by dragging fish to feed, so I'd recommend taking an icthyornis (seagull) before attempting to tame even a minimum level, since their golden fish buff is absolutely needed.
Once she's tamed, find a pond with a lot of fish in it; try to make sure it's safe for your goose! Throw her into the water and set her to wander and aggressive. Then stand back and let her do her work.
Once your hesperornis starts to appear passive, even in the presence of coel, she's ready! Pick her up on your shoulder and throw her on shore, and she'll lay an egg and return to the water on her own. This is usually a regular egg, but don't worry, since these can be refined into oil in your cooking pot!
Once you do get a golden egg, make sure you're ready to use it (have Laz Chowder premade, if you're making kibble!) cause these things spoil and they spoil FAST.
Tamed Hesperornis WILL lay golden eggs, just takes a bit of time. I had mine on follow in the Redwoods river, and whistle attack targeted it on a fish. It must kill and harvest SEVERAL fish before going on land to lay an egg. It will either lay a normal duck egg, or it will lay the golden egg. It seems to be random, however mine laid about five normal eggs for every golden egg. If the type and size of fish matters, I have not figured that out yet. It will usually kill all types in one shot, and it will not agro every single sabertooth Salmon in the vicinity when it kills one so don't worry about them ganging up on your duck. I was using a lvl 80, probably tamed around level 50 or so and all I put points in was health. Have it kill every single fish in the river before you give up trying for that golden egg. It will happen, give it time. Hope this helps :)