Update:
I just purchased a Logitech Keys keyboard, and a thunderbolt to HDMI adapter.
Connected both to my MBA 2015, and I can close the lid, and use the laptop normally (connected to my Samsung monitor). No app install or anything 🙂
PS: The adapter is the Microsoft one, the one normally used for the Surface. Not sure if that is why the MBA works with the lid shut (active vs inactive adapters).
You know, I should reboot the MBA, to see if this remains as is.
Now for some cable management, since the HDMI port is on the bloody opposite side of the power connector 🙁