Is there a Prophetic Hadith that encourages hunting—of all types—in order to defy the Jews?

There is no authentic hadith that encourages hunting birds. Rather, there is a narration that prohibits it in the hadith: "Whoever lives in the desert becomes coarse, and whoever follows game becomes heedless." Ibn Hajar used this as evidence for the dislike of excessive hunting, because it may distract one from obligatory and recommended acts.