Originally Posted by evoG

Slightly faster than brute forcing all possible JFK-DXB by DXB-ICN permutations would be to find EK's north american gateway with the most availability -DXB (or MXP) near your departure dates (may or may not be JFK), then use the award calender for that gateway-ICN. This way you can search a month at a time and by picking the least-hindered gateway you've allowed more DXB-ICN to be displayed.

For more comprehensive searching, find some date with JFK-DXB availability several days in advance of your earliest desired date for DXB-ICN, then use multi city search to feel out availability for the ICN segment day by day. This way you know that the -DXB segment isn't the bottleneck. The multi city search displays results a week at a time and that may provide additional clues and speed up your search.