Blue Bandana

Blue Bandana speeds up machines by 7.5% of your Extraction Speed, and charges you 5% of your Skill Check chance for it. With 45 units per Machine, a 1-star extractor goes from 60 seconds to about 56, a 3-star from 45 to about 41.9, and a 5-star from 30 to about 28. The wiki points out a quirk: because Skill Checks also add progress, losing 5% of their chance means this is the only Trinket that can make a Machine slower for certain stat distributions.

It therefore suits Toons that do not rely on Skill Checks for speed, or that already have a generous window. The ones with the slowest passive filling, Pebble, Razzle & Dazzle and Shrimpo at 1 star of Extraction Speed, save about 4 seconds on every Machine, which adds up over a 20-Machine floor. Boxten, the starter Toon whose Twisted unlocks it, has 3 stars in everything.

Twisted Boxten is a Common (75 of 110 spawn weight at the start), so the Research is among the fastest to finish. There is no Ichor price. The wiki adds that this Trinket used to raise Skill Check chance by 5% before version 0.2 flipped it. It is available now.