Claire is right on. Shown is a 3/8" deep hole (gun) drill made from a 36" length of drill rod. This one has an extension to 44" for drilling ramrod holes. I've been using it for decades.
This is the cutting end.
And this is the 3/4" counterbore you need to go with the drill. This is with a 1/4" semi-floating pilot, I also have a 3/8" pilot.
I through drill with the lathe between centers. Both ends of the stock blank need to be perpendicular to the hole before starting. I usually drill half way from each end.
The counterboring is done with a drill motor and an extension.
Steve
BTW, I'd think about drilling the hole first, then fitting the transplant to align with the existing hole. You might be able to use a draw bolt to suck the two together for fitting.