Would you use a drill bit extension for a Forstner bit?
I make quite a few mills per year and that's exactly what I do.
I drill the clearance holes, the ones which house the actual grinding mechanism, in the base of the mill first, to whatever depth is required by the mechanism I've chosen, then the through hole last of all.
(btw, a good way to gauge hole depth in real time is to do the math with the thread rate of your tailstock quill then count the revolutions of the handwheel from the moment the drill bit touches the wood — mine is 12 threads per inch, so six revolutions of the handwheel is exactly 1/2", 3 would be 1/4", and 1 1/2 revolutions would be 1/8" and so on. Much quicker than actually measuring the depth with anything, and really quite accurate. I did all the math for any measurement from 0 to 1", to the nearest 1/16", and wrote them all down on a sheet that's pinned to the wall where I can see it. That way I don't have to remember anything, and I don't have to measure anything either.)
I drill the through hole in two steps, roughly halfway through from one end of the mill blank, then reverse the blank, center it carefully, and then come in from the other end until I meet the first hole.
That way if there is any slight discrepancy due to the drill wandering off center a little bit, it's in the middle of peppermill and nobody will see it.
Plus, I can drill a 10+" mill blank easily with only a 6" extension.