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.
I was starting out and I killed one of these because they would attacking me then I saw two and tried to tame the female which I did but then it just deleted my save (one tip if you don’t want it to hurt you have a fish in your hands like your carrying a fish)
The most frustrating thing about taming these guys is not only do they take a lot of fish but the same button to feed them is the same button to drop the fish(i don’t know if it’s different in pc or ps4)and it’s constantly swinging around your face making it annoying because you’re constantly dropping it. Either make it so the controls to feed them are different then dropping it or make ur so it’s not constantly in ur face.
Tame to this kill fish and drag there body to it and once tamed if you feed them more dead fish corpses it will build up and lay a golden egg which is used as a buff to certain dinos or oil or a player buff
What I did was use my Ichthyornis to Hunt fishes and bring the dead bodies for me.
Change his settings to "Hunt and retrieve", put him on your shoulder then you go swim. While you are swimming aim at The fish you want and double tap F (on PC). Ichthyornis will fly, kill the fish and bring the body to you.
Then you take the body to Hesperornis and feed it. Once you get used to it it's very fast and easy. I recommend following Hesp from behind so you dont need to adjust your aim all the time.
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 :)
When you anger it. Don’t try to laugh because you will laugh. It looks even more pathetic than a dodo rushing at you and it deals LITERALLY 1 damage. Upgrading it to 200% means it only deals like 2. Don’t take it to fight.