do you have more pics of the sole? Looks like double leather /Cuban heel /then the mini lug sole. Do they automatically add another layer of leather when adding the lug sole? I know whites add another layer when they do commando soles so wondering if they do the same with half soles. Post your spec
Sorry for just getting to respond to this. Pretty much all of White's soles are the same thickness if I'm not mistaken, so for the mini lug half sole they add a second piece of leather to make up the difference. It gives the appearance of a double leather with the half sole beneath it. Here is a better picture of it in action.
I am waiting for a brown edge dressing I ordered from AE (thanks to those who suggested it) so I will retouch them shortly.