Speedrun the event track

Completing an assignment in Vanguard progresses the "Complete 6 Vanguard expeditions" task by one point.
There are 2 Vanguard Expeditions which have very easy/fast assignments.

Log in all 3 EVE characters on the account you want to progress the EVE Online event track for, BEFORE you do the assignments in EVE Vanguard. Open the event track with each character. After that, the EVE client can be closed. This ensures that all 3 characters can be progressed at the same time!

To begin, accept the "Avalon Rises" Expedition in EVE Vanguard and activate the first assignment which requires you to kill 3 NPCs and loot 6 boxes (looting empty boxes as well panels works too). Once the assignment is done, extract/die and complete assignment in the warbarge. Now ABANDON* the expedition and accept it again. Rinse repeat until you did it 6 times. Then log in to the EVE Online characters to claim the task. Do that 4 times in total and all 3 EVE characters will have completed the event.

There's a secondary Expedition that can help speed this up even more. Looting the boxes may award "Intel Fragment", "Insulated Cabling", "Silicate Circuitry" and "Flux Coil". Expedition "Beneath The Garden" has two assignments for turning in these.

A third Expedition called "The Wager" let's you turn in Sec-5 Command Implants which drop from commanders. If you happen to kill some, this will help speed-run this too.

Item name	Icon
Intel fragment visual
Insulated Cabling
Silicate Circuitry
Flux Coil

Abandon button location:

Thanks to Cyric Cerica, Ben Fraction and everybody else who helped putting this article together!