How do I get to the "Fowl Lair"

To get to the Fowl Lair you need to use the Ancient Device in Desolate Sands. It is one of the 10 possible outcomes of using the Ancient Device. So take note that not only is the spawn of the device random but there's only 10% of getting the Fowl Lair.

Where are all the random dungeons?

This contains a community wiki of random locations and which zones they are in. It was revised to add Fowl Lair based on this forum thread:

http://us.battle.net/d3/en/forum/topic/5149619068

The plateau on the map (obviously random location, but that's the shape)

The Ancient Device

The entrance to the Fowl Lair after succesfully spawning it using the device (took A LOT of tries)

Inside the Fowl Lair