Most efficient in terms of time spent is probably doing Mythic ABT once a week (you're doing that anyways), doing WQs for the emissary caches and doing class hall missions at 200% chance (these take almost no time to queue up). Not sure about mythic dungeons, to be honest - have never paid attention if they actually drop essences. If they do, then do them all once a week too (in order from shortest to longest).
Fastest in terms of getting the legendaries you want in the shortest amount of time possible will also include all world quests (currently the WQs I have available add up to 200 essences, this includes fully unlocked Argus though), and legion invasion WQs whenever they're up. Not sure if your first random daily heroic dungeon of the week will reward some on your 110 (111s can't do random legion dungeons).
All in all, you can easily get 1 legendary per week if you do ABT and do every other activity listed. Given that to actually gear your twink to adequate levels will take longer than a couple of weeks, it's not something you should particularly lose any sleep over.
If you played actively in Legion, you might have 1000s left on your old main. You can buy BoA class-specific tokens for 1000 essences and mail to your twinks (though you'll be playing the lottery on actually getting the legendary you want). Remember those tokens make legendaries based on your loot spec.
