Empire Hunts are activities that appeared in Beyond Light expansion. In them, you have to track down a special Fallen enemies throughout the Europa. They are unlocked by finishing missions for Variks the Loyal.
In total, there are 3 Empire Hunts: The Warrior, The Dark Priestess, and The Technocrat, each of which finishes with a boss fight. Empire Hunts are replayable after you finish the Reclaiming Europa quest from Zavala. You initiate them via Variks for 40 Herealways Pieces, which are a special currency you acquire by finishing Patrols, Lost Sectors, and public events.