For more Hesperornis taming details, get the taming calculator app.
To tame a Hesperornis, you must kill a fish, and drag the fish's corpse to the Hesperornis to feed it. The fish's size plays a major factor in how many fish will be required (a 1.0x fish appears to fill up the taming bar twice as much as a 0.5x fish). Level of fish does not appear to impact taming. The Sabertooth Salmon is by far the most effective fish, equivalent of about 4 Coels of the same size, while the Piranha is equivalent to about 2.4 Coels.
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.