There are many cool routings, but since you are traveling in summer, I think you should check for availability first before consider the routings.
OZ F JFK-ICN-HKG
NH F JFK-NRT-HKG C for NRT-HKG
SQ forget about it, you can't redemm F or C on 77W, and LAX-NRT though it is 744, the connection time in NRT does not have enough time for you to make it on NH NRT-HKG.
UA PS F JFK-LAX/SFO, then NH LAX/SFO-NRT or OZ LAX/SFO-ICN
Note (1): OZ and NH C is angle flat seat except NH JFK-NRT.
Note (2): OZ has numerous blackout dates in summer months, when the dates are not available, it is most likely it falls on one of those dates instead of seats are taken
Originating from PHL and in F/C, you can do PHL-FRA-HKG on LH, especially in F for the F terminal. If originating from JFK, you can do JFK-FRA on either LH or your beloved SQ, then FRA-HKG on LH. I am not sure if LX flies to HKG from either ZRH or GVA, if so LX should be good in the premium cabins. Other available options out of JFK (namely, UA, NH, & SQ over the pacific) have already be mentioned except for OZ, which is pretty good. If you are willing to make another connection, that will open up more routing possibilities. Hope this helps.
Routings thru Atlantic are good, but that would mean more miles.
LH so far is good as long as it is not a 744. 744 F upstairs are not ideal, having seatmates are bad idea for F in the first place. LX I am not sure if A340 has new F seats yet. From the trip reports it looks ok, but not the kind of experience you will see on SQ F, NH F, or OZ F.
There is also a third option, it is NZ C LHR-HKG