I started the watch and waited until 10 seconds to attack the boss. I got him down to about 11% and made sure he had Flame Shock on him.
Then I dropped down to the nest and killed the 2 chicks. After that I flew up just in time to have the Golden Egg drop on my head. Then I used gust of wind to give me more air time with the egg and keep the DoTs ticking. While I didn't make the platform, the extra time allowed the Fire Elemental to kill the boss in time. It's definitely possible to land on the platform with Gust of Wind however if you mess up the DoTs.