So, the actual Breville ball valves are NLA (no longer available) with the exception of a tip that Forum Home Appliances in Canada has a few left, (and I ordered one on a kind tip from someone over in the Breville Dual Boiler (BDB), the end of the line? thread).
Anyway, all is not necessarily lost if we can't get the Breville part any more. As with many sealing and fluid control elements on the BDB, they are essentially off the shelf-type parts. Ball valves should be no exception. The world is awash in ball valves and it's certain that dozens of them out there will do the job, some requiring more "fitting" than others. It will be incumbent on the BDB community to help find them.
Here's what I know so far:
This is the ball valve picture from ereplacementparts, that is listed as "discontinued":

The middle section is the actual ball valve. The parts screwed into the ends are the Breville specific pieces. You should always keep those.
For redundancy, here is a picture of an old leaky ball valve given to me kindly by Klund, a valuable contributor to the BDB community:
Here is the same ball valve with the Breville parts unscrewed from the actual ball valve section:
The threads are BSPP (British) G1/4 thread. Female-Female ball valves with this thread are available online. Also, USA 1/4 NPT thread fits, and will probably work. Definitely use teflon tape to seal threads. The top and bottom male threaded fitting are the parts you keep. The middle section is the one you replace if you are speccing your own ball valve.
Here is a ball valve I got off eBay with the same g1/4 British thread as the original piece. As you can see, the Breville parts on the ends fit it perfectly. It's a wee bit taller and bigger around the middle than the original ball valve, but with some fabbing, it ought to work:
Not sure when I'll get around to trying the alternate ball valve, but I'll report back when I do.
Venture forth, BDB'ers and let's get this done...
-Peter