Depends on what you mean by 'Church of the East', but I believe an answer I wrote on when Catholicism developed may answer your question, since it discusses how there came to be 'west' churches and 'east' churches.

https://www.reddit.com/r/AskHistorians/comments/9hxndy/from_an_historical_perspective_how_old_is_the/e6fq579/