During the battle with Gulp, eggs will be dropped into the arena from the air. Inside these eggs are different "weapons" and sometimes chickens.
You need to avoid charging at the eggs/chickens while also trying to avoid flaming them.
In my experience, if you avoid being hit at all and keep full health, no chickens will ever enter the level so Gulp can't eat them to up his health
Easy way [confirmed]
*** Spoiler - click to reveal ***At the end of the game you can unlock permanent super flame, so coming back to this level with it will make this achievement much easier as you can just flame him quickly