What is your budget and what class are you looking at flying in?
If economy then I would suggest just using Skyscanner and seeing which date / airline is cheapest and not be too concerned about connections. Likely it will be Hong Kong or Bangkok.
If you want to fly with a bit more luxury, but don't want to pay full Business / First prices, consider flying ex-EU (a flight that originates from outside of the UK) with British Airways. Even with an initial positioning flight it still works out far far cheaper than a direct LON - XYZ flight. What's better, is if you have any Avios, or are enrolled in an American Express MR scheme, your positioning flights will be a mere £35 each and you get to tick another city off of your travel list (you said you have spare time, hence why I am suggesting this option).
Cheapest economy (no lie-flat beds) for the start of Feb (flying LON (LHR or LGW) to HAN return) seems to be:
£912 Air France
£2,784 Alitalia
£1,212 ANA
£542 Asiana Airlines
£664 Austrian Airlines
£647 British Airways
£795 Cathay Pacific
£609 China Southern
£541 Etihad Airways
£1,426 Japan Airlines
£952 KLM
£799 Korean Air
£608 Malaysia Airlines
£543 Qatar Airways
£760 Singapore Airlines
£824 Thai Airways
£617 Vietnam Airlines
£616 Airline combinations
Business class from LON - HAN return:
£2,560 Air France
£2,897 Asiana Airlines
£1,833 Austrian Airlines
£3,396 British Airways
£3,522 Cathay Pacific
£2,190 China Southern
£2,929 Etihad Airways
£3,731 Japan Airlines
£2,570 KLM
£2,640 Korean Air
£2,336 Malaysia Airlines
£2,307 Qatar Airways
£3,237 Singapore Airlines
£2,701 Thai Airways
£2,385 Vietnam Airlines
£2,221 Airline combinations
BA for the above route would be in £3,400 - but due to the attractive pricing structure they have to lure in European customers, you can fly Business Class with them out from AMS (Amsterdam) and back to ATH (Athens) and save £1,200 per person. The total flight cost of doing this would be £2,238 each. The only cheaper business option would be China Southern, and I can't say their business class product would match up to British Airways.