Odyalutai Hawk
Thanks to Nexus Mods - 'Companion Unlock' I caught this by building stairs near where they are flying in a group, then in Build menu I chose Bio-Dome, clicked 'V' for Toggle Free Placement, and then clicked 'B' for Toggle Camera. Took a few tries to watch one fly into the Bio-Dome and click to create it with this bird inside, flapping its wings. I then jumped on top of Dome, and free placed wood flooring near the edge to allow building a doorway. Walking inside, it was easy to make it my pet as usual with pellets, made it happy and well fed, and created a save point. Jumped to the ground and called it to me for this flight. When fed, create eggs is instantly available (more than 1 if fed again); They are 'hungry' right away after 'laying an egg' :-) Using NMS Editor (available free on github)- it let me copy the seeds easily to another save, tested fine and flew too. Refiner Duplicating Glitch does NOT work, as the Eggs created this way are not valid. Here's good Reddit link that showed me how: https://www.reddit.com/r/NoMansSkyTheGame/s/BeXPtAEtbX - Glyphs at end of video.