Brief Custody
Kill a guardian within 12 seconds of first striking it.

The 12 seconds start at the guardian's first damage from anything: your cards, a burn, your grave-warden, a zone you placed or a trap. Only guardians count, the holders of the way down on floors that are not a multiple of 5. A boss does not count.

Do it on depth 1

Go for the Warden Stairkeeper on depth 1. It holds the portal in the chamber farthest from where you arrive, and it rolls only two cards. Enemies gain 26% health for every floor below the first, so the first floor's guardian is the easiest to kill quickly.
Clear its escort first. A single-target card strikes the nearest enemy in front of you, so others near the guardian draw those casts. Deal with them before the guardian takes its first hit.
Don't start the clock early. A burn left ticking or a placed zone it walks through counts as the first strike. Once it is hit, stay on it: your cards cast by themselves at whatever is in front of you, so keep it in front.
The card comes up when it dies: DEED EARNED · Brief Custody.

Depth 1, TIME 0:25: the hero fighting the guardian, its health bar down to a quarter — Depth 1: the guardian's health bar, top centre, falling.

Depth 1, TIME 0:26: the Brief Custody card, deed earned — The guardian falls: Brief Custody.

The recorded run

One run from depth 1 to depth 6, played by the game's test autopilot at the game's real speed on an account with every Forge track at rank 12. It earned all five speed achievements: Brief Custody, Three Stairs at a Run, Grave-Robber's Hours, Smash and Grab and No Time for Grave-Goods. Brief Custody came on depth 1, 40 seconds in.

The HUD clock in this video reads low. The test autopilot stops the run clock whenever it halts to fight, so its TIME shows 1:13 on arriving at depth 4, after 1:44 of play. A player's clock keeps running through fights. The floor times and the guardian's 12 seconds are counted on a separate clock, which is not affected.

The DEEDS screen of the same account: Brief Custody held — The same account's DEEDS screen afterwards: Brief Custody held.

What goes wrong

A slow opening kit. On a fresh account you start with one card from the opening offer. Take the first-blood offer's card, and anything else that adds damage, before you open the guardian's chamber.
Chipping it and backing off. The count does not pause while you retreat. If it has survived 12 seconds, that guardian is spent; try the next floor's.
Depths 2 to 4 have the same guardian, a little tougher each floor. From depth 6 it is the Household Doorward.