OP never stated time frames for travel, so my suggestion would be:

CX JFK-HKG, perhaps a layover in a hectic city, ot just spending time in top CX Lounges Pier or Wing, OW status or ticket permitting.
Then CX HKG- european hub; LHR can be a drag, but BA lounges sweetens you up after all the hustle, FRA has quick connections to CPH, but carriers FRA-CPH has little to offer when it comes to lounges. FMC and CDG about the same and AMS somewhere intermediate.

MY Choice would be JFK-HKG-LHR-CPH (last leg BA, CX codeshare)

To fly the shortest way might be considered a heresy