Interesting question.
Generally a badge die is ground out to roughly the design required before being hobbed. Basically you have to push the metal somewhere so removing as much as possible beforehand is preferable .
A lot will depend on production numbers so making an entirely new die isn't unlikely. It's easy to underestimate the skill of hand die makers as they would be doing this day in and day out, so making a near identical revision isn't difficult. You take all your layout measures off the first.
The centres may have been seperate punches applied afterwards, hence they often tend to show a very untidy style of join that looks like a soldering fillet.
Another aspect oftem overlooked can be the use of copy lathes (essentially a giant pantograph, commonly used in coining) working from plaster models three or four times life size. These can serve to quickly grind out the badge recess before hobbing them or even produce dies directly.
Much of the die makers skills comes from the polishing fine cutting applied afterwards from cleaning up in the post hardening phase. A lot is done with small stone slips and emery powder with a wooden peg.
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