FWIW, the how-to starts on p.33 of the Baker thesis.
This is all done at very high pressure---the relief valve in the proof-of-principle apparatus is rated at 1420 PSI. No obscure cryogenic materials seem to be used, just stainless steel.
Really, though, this isn't in the realm of a garage project unless you know what you're doing. You've got to be maintaining high pressures throughout the system, including the ingredient and product containers, and you're dealing with liquids that are intended to freeze solid. That introduces opportunities for things to clog, running the risk of vents failing to vent, which could lead to catastrophic (deadly) failures. This is far more involved than offgassing CO2 into a keg of beer.
If you can get your parts machined and/or buy parts intended for this kind of use, that's one thing. But I would really caution against kludging something together by the seat of your pants on this.