Disclaimer: this list was created for my personal reference. Farm strats may be suboptimal; fragments I don't use frequently may be missing.
Forgotten Fragment of Skill: Normal mobs in the south zone of Bozja. Easy farm: 1x tank + 1x DPS kill rank iii matamatas south of Olana's Stand aetheryte. Yield: 2-3 fragments per mob.
Forgotten Fragment of Mastery: Battle for Castrum Lacus Litore bonus drop from saving enough prisoners. Drops Deep versions of Skill essences.
Forgotten Fragment of Awakening: Normal mobs in the central zone of Bozja. Easy farm: 1x bloodsucker tank (+ optional tank defense banner) + dps for speed. Kill rank iii antlions near center of middle zone. Yield: ~1 fragment per mob.
Forgotten Fragment of Becoming: Battle for Castrum Lacus Litore bonus drop from saving enough prisoners. Drops Deep versions of Awakening essences.
Forgotten Fragment of Superstition: Battle for Castrum Lacus Litore bonus drop from saving enough prisoners.
Forgotten Fragment of Transcendence: Battle for Castrum Lacus Litore bonus drop from saving enough prisoners. Drops Deep versions of Superstition essences.
Forgotten Fragment of Caution: Star mobs in the south zone of Bozja. Easy farm: Lost Death.
Forgotten Fragment of Ingenuity: Star mobs in the middle zone of Bozja. Easy farm: Lost Death.
Forgotten Fragment of Violence: Star mobs in the north zone of Bozja. Easy farm: Lost Death.
Forgotten Fragment of Compassion: Normal mobs in the north zone of Bozja. More science needed for farming.
Forgotten Fragment of Support: Sprites in the north zone of Bozja and Zadnor. Easy farm: reflecting wind sprites in Bozja (rank v, south of Camp Steva aetheryte) or earth sprites in Bozja (rank iv, middle south edge of zone), or ice sprites in Zadnor (rank v, north of Hrmovr Point aetheryte) or wind sprites in Zadnor (rank iv, center north edge of zone).
Forgotten Fragment of Resolve: Looted from Bozja Critical Engagements: Trampled Under Hoof, Where Strode the Behemoth, Metal Fox Chaos.
Forgotten Fragment of Sagacity: Battle for Castrum Lacus Litore bonus drop from saving enough prisoners.
Forgotten Fragment of Artistry: Star mobs and sprites in the west zone of Zadnor. Easy farm: Lost Death, or reflecting water sprites (rank v, northwest edge of zone) or lightning sprites (rank iv, east edge of smallest island).
Forgotten Fragment of Desperation: Normal mobs in the north zone of Zadnor. More science needed for farming.
Forgotten Fragment of Tenacity: Looted from specific Critical Engagements in Zadnor. Easy farm: Marketboard! These are usually cheap.
Forgotten Fragment of Preparation: Sprites and ashkin in the south zone of Bozja. Easy farm: reflecting lightning sprites (rank v, southeast corner of zone) or water sprites (rank iv, south of Olana's Stand aetheryte).
Forgotten Fragment of Care: Sprites and ashkin in the middle zone of Bozja. Easy farm: reflecting earth sprites (rank v, northwest corner of zone) or wind sprites (rank iv, middle north edge of zone).
Forgotten Fragment of History: Star mobs and sprites in the south zone of Zadnor. Easy farm: Lost Death, or reflecting wind sprites (rank v, western edge of zone) or ice sprites (outside north exit of cave area).
Forgotten Fragment of Moonlight: Normal mobs in the middle zone of Zadnor. More science needed for farm.
All five Pure Essence fragments (not including Deception) are looted from Delubrum Reginae, but can also be purchased in exchange for Bozjan clusters at a rate of 1 cluster to 1 fragment. Each fragment has an even chance of dropping a Pure Essence or a powerful action. Pure essences can ONLY be used in Delubrum Reginae (either difficulty). Easy farm: Bozjan cluster farming.
Pure Essence of the Gambler is a very niche essence with strange RNG-based effects. It drops from Forgotten Fragments of Deception, which are only obtainable in Delubrum Reginae (Savage).