Kill the "chicken"

 is an unmarked quest in Fallout 2.

Walkthrough
The "chicken" that Rose gets her Wasteland Omlettes from is actually a very fearsome Deathclaw!

You can kill it for 1000 quest XP and a Modoc town rep drop. Rose also won't be able to sell Omlettes.

The quest is mentioned in Fallout: New Vegas in the small mining town of Sloan. The local Chef describes Rose as her great-grandmother and mentions someone killed her Deathclaw by "shooting it in the eyes."