Instructions
Travel to Mulgore. The entrance to Mulgore sits east of the Southern Barrens. If you're alliance, the easiest way to get there is by taking a boat to Ratchet in the Barrens. If you're horde, take a zeppelin to Orgrimmar.
Patrol central to northern Mulgore and look for wiry swoops and taloned swoops, both of which are flying birds. Wiry swoops appear near the center of Mulgore, while taloned swoops appear to the north.
Kill the swoops when you see them. Right-click their corpses to look inside them. You will occasionally find a small egg.