Could someone explain the method for cutting these very fine threads on large diameter pieces?
An interesting topic that is occasionally required, but seldom if ever addressed in Horological repair. (At least that I have seen)
However, you have not clarified if for wrist or pocket watches and if for new construction or cleanup/ restoration of existing threads. As such, I will touch on all.
As others have mentioned, you will need a lathe capable of single point cutting the thread size required. With this capability, you can either machine Taps and Dies or single point thread the work piece directly. With new construction, you can start with the work piece in a shape easily mounted in a Lathe chuck for single point threading. While most existing wrist case backs can also be held this way, existing pocket watch bezels and lids are often another story. In this case, the cases were/are shaped with the lids and bezels screwed in place.
Again as such, the sharp edged examples are seldom perfectly round nor is the thread perfectly centered to the OD. For this issue, I have installed four aluminum posts in machinable jaws with a ID "V" machined to hold the sharp edge without damage per attached photo. In addition, I can rotate the bezel or lid until I find a spot where the ID thread runs true.
Considering the short thread requiring that the thread terminate in an exact spot for clearance, I would suggest a hand cranked spindle utilizing a lead screw carriage. In addition, the carriage hand wheel should contain clear and precise calibration markings that can be utilized like a micrometer for depth control. In this case, when setting the threading tool for both new and existing threads, the tool tip is positioned at the desired depth. The hand wheel is then zeroed and threading stops each time the carriage hand wheel registers zero and the tool is retracted.
When setting up a Lathe for small threads, a slight amount of backlash is required in both gear train and lead screw to permit free operation of the system. Cleaning or restoring an existing damaged thread of this size requires exact positioning of the cutting to tip that is a moving target when backlash exists. To eliminate backlash, I spring load the carriage per the second attached photo. This also gives a more accurate thread on small new cut threads. On existing threads, the threading tool tip is set in an existing thread from the rear of the lathe looking down under optics. To assist precise positioning, a fine threaded setscrew was added behind the threading tool that can be seen in both photo`s
Jerry Kieffer