@Ati
Indeed we have to try and logically figure out a base starting point for those Eggs and why Dr Shaw was his greatest Specimen, all we see from Dr Shaw's Corpse is she has had her Chest Cavity removed with some Organs missing... Its likely that she was Face Hugged and David then removed something from her.
The Question would be was this Organism what leads to those Eggs? If so why did David have to get rid of this Organism as its not alive when the Covenant arrives, maybe its one of the Specimens in his Work Shop?
Then also it could be that David needed Dr Shaws Reproductive Organs?
I think a good way to look at this would be to completely Scrap Alien Covenant... imagine it never happened..
Look at being David, with Dr Shaw and the Black Goo and how David maybe never got to see the final results of the experiment he did on Holloway.
We had the Trilobite a kind of Face Hugger, which would have been the result of one of 3 ways.
1) Holloways infected Xeno-Virus Sperm, Fertilizes Shaws Egg Cell
2) Holloway merely passed on tiny nano-Xeno Virus worms that was transmitted to Dr Shaw and one of these infected her Egg Cell.
3) As Per 2) but instead of infecting a Egg Cell, the Nano Worm just grew inside her Womb.
I always felt looking at the clues that option 2) was the most likely case.
So if you was David you could repeat the Process, by simply infecting Dr Shaws Egg Cells with the Black Goo.
I assume that its Dr Shaws Egg Cells that provided the Foundation of Davids Experiments and Once he had grown a Specimen he then introduced it to its Mother... and thus Dr Shaw gestated a earlier version of Davids Creation.