In a world where chickens are apparently extinct, it seems some survived underground and lead short lives on the surface protected by the short-lived residents of the Hilltop Farm ruins. Miles was most likely the primary caretaker as the only one with farm experience.
Chickens are gregarious, domesticated fowl reared for meat and eggs. They are the most common and widespread domesticated animals whose ancestors were domesticated up to 8000 years ago for food and/or entertainment in the form of cockfighting.
There are a few minor references to chicken in the Fallout series, for example Armitage in Fallout 3 claims he could "break your neck like a chicken bone", and Mom in Fallout 2 referring to radscorpion en Croute as tasting "like chicken", meaning that knowledge, and even the taste, of chicken can be recalled by some after the Great War.
Actual chickens are referred to in Fallout 2, Fallout 3 as taste and Fallout Tactics in the form of graffiti. Others may be found but only as a means of calling someone afraid. Fallout 4's Far Harbor DLC is the only instance in the entire fallout universe to date that features live chickens.