The time difference is closely related to how many lines of longitude you cross. So you're looking for as many crossed as possible, in the shortest time.
The time of a flight is closely related to its distance (for any given groundspeed), and is reduced if the prevailing air movement is from behind it (because airspeed is related to groundspeed plus speed of air movement/currents).
So you want to cross as many lines of longitude as possible, in the shortest possible distance, with the wind coming from behind. That immediately suggests you want to be travelling as close as possible to the north or south poles, and in a west-to-east direction for northern extremes, or east-to-west for southern extremes.
You are limited because too close to the poles,you lose time zones - nobody bothers with time zones being 12 hours different if you and a friend both start at the north pole and walk a mile in opposite directions. But the Arctic is almost certainly better for this than the Antarctic, because many countries are closer, with more interest in asserting territory (hence time zones), and longer explored/less isolated.
Your air flights at high latitudes are likely to be from ad hoc runways. You wont find big towns there. But even small communities may have local runways, and your question doesn't exclude those.
Putting this all together, I think your answer won't be found by looking at major airline schedules.
You need to find where in the arctic you could find the most northerly territories which have time zones. Then look for the most northernly settlements in those areas. Then, with a map centred on the true or magnetic north pole (whichever longitude originates at), figure out the shortest distance between settlements as far around the "clock face" as you can.
You'll have to do this manually, because there are unlikely to be scheduled flights, but that's how you'd do it. I suspect your answer would involve Arctic airbases or exploratory stations belonging to Russia and the US.
If you flew the path you found, you'd get maximum time zones per flying hour.