The Pirate Dragon Egg Fragment (a.k.a. Essence of Flakbeard) is a rare crafting material that is dropped by the world boss, Flakbeard. These fragments are used to craft Flakbeard, the Relentless.
Flakbeard can be found in any Treasure Isles biome in any of the Uber Worlds above Uber-2. The boss can spawn on land, or in their dungeon, which looks green and can be found on the floor of a Treasure Isles biome.
The amount of fragments dropped depends on the Uber world the player kills Flakbeard in. The higher the Uber, the more fragments drop.