The Victorian to Edwardian periods - the early Poultry Fancy, and 1890s-1920 high-efficiency-breed development period - shared an interesting aspect: a lot of the new breeds, and breed refining that went on, used genes from newly-available Asian fowl. The Modern era could also be called the Melting Pot era.
I do think Modern has a ring to it, especially as people from this time period were acutely conscious of their own modernity, and of "modern scientific methods" in poultry breeding and rearing, but I understand Yellow House Farm's objection.
Post Modern probably should be knocked out of the running (didn't mean it as a serious suggestion myself)... The term in art, literature and architecture is associated with merciless tongue-in-cheek quoting and mixing-up of disparate time periods and aesthetics, eg. an office building shaped like a huge thatched cottage made of metal and glass. A post-modern chicken sounds to me like a whimsical creature made up of exaggerated elements drawn from other, traditional and world breeds... and now my head hurts, as I think I may have just described the Faverolles.
Best - exop