Can you build a building for free with the ability of the Egyptians and the n in the same city spend your production using the engineering technology(spliting production to buy)?

Thanks!

No. You get 1 action per city during the City Management phase and the Egyptians can build for free as an action (you dont spend production to build it so Engineering does not affect the Egyptians ability). So your action during the City Management phase is either build at a cost (using Engineering to split the production to build 2 buildings) or build for free.