The Falconeer

Mission 1 of chapter 2 is a good place for this.

Before you start, run the Seaenchantress side-mission from Dunkle for 1000 splinters at a time. Do this to ensure you have enough to purchase the 3 star heavy rifle from the Dunkle tradesman, it will cost you 3500 splinters.

Once equipped, simply do this mission. You are tasked with finding a lost ship/shipment and it will take you to three separate locations. Along the way you will encounter enemy aircraft which resemble giant flies. On easy mode, with the heavy rifle equipped, they go down in three hits. I was able to knock out 12 of them in this mission with little to no effort.

The achievement requires you to be on a mission while taking down 15 (10 is a typo in the solution, as confirmed by the game's developer; will be fixed to 10 in an upcoming patch) air targets.

You can be on a main or a side mission to have air targets count towards this achievement, but they must all be downed in the same mission. Side-missions are easier because you won't have company that can "steal" air targets from you. For this achievement, I recommend starting a side-mission in chapter 3 at Mawbridge that takes you south or east. Fly a pattern between Basilicus Primus (just south of the eastern edge of the maw), Stargazer (south-south west of Basilicus Primus) and Oberon's Reach (south-west of Stargazer). Each of these locations holds 2 enemy wyvern you can down for this achievement.

You can of course down other targets as well, but only air targets netting splinters on their demise will count towards this achievement, hence, air ships do not count.

While you are doing this, there are 2 Turrets at Basilicus Primus (and 2 more nearby in the Maw) and 4 at Stargazer should you want to farm other achievements as well. There is also always one boat spawn at Stargazer.

I unlocked Ace of Aces in the third chapter, I started a side mission, completed the mission's objectives, and once the mission complete message flashed up (usually at this point you'd need to return to the Shard) and I headed over to the Stargazer (this is an enemy faction in chapter 3) and here there is a large airship that spawns.

Once I've destroyed it I flew to Oberan Reach which also had another airship. Once I destroyed the airship at Oberan Reach, I went back to Stargazer, then rinse and repeat until I counted around 12 (just to be safe!). I then handed in the mission and the achievement popped.

Hope it helps!