Mod Marv's Lantern vise

[Toolmaker51]

What about something like this critter: https://smile.amazon.com/M10-1-25-Sp...split+hex+dies
I've never seen these before, but "hex split die" search brought it up. However, they might be rather costly if you needed a whole set of them.

Not sure how this gets named chaser. Chasing threads [a different activity than we pursue @ HMT.net] is a repair technique to clean root, re-form crest, and shave flanks so it will enter again, not necessarily perfect per original class of fit.
Cutting dies produce threads, usually in a single pass, manner identical to lathes. That accounts for the leading side being chamfered. The adjustment [range is limited] can be used to adjust clearance [pitch diameter mainly] of desired class of fit. It also allows a roughing pass on tough material. You'd want a thread mike, wires, or go/no-go handy to monitor changes.