Actually, a better name would be “The limited-water shave”: how to shave when you have only a small amount of heated water—as when camping, for example. But the most common use of this shaving technique is for shaving when the tap water is insufferably hard and thus cannot produce good lather from a soap because the minerals in the water immediately bond to the dissolved soap, forming a sticky scum that clings to bathtub, shower walls, sink, and your skin.
A shaver who lives in the same town where he was raised may not even realize that the water is hard—it’s just water, the same stuff he’s used all his life. He may note that he can’t get good lather from soaps, but shaving creams do fine (because they are less affected by hard water), so he figures he must have “shaving-cream brushes” rather than “soap brushes”. But the problem is the water, not the brushes.
The distilled-water shave is both a way to test your tap water through a performance comparison, and also it’s a workaround in case installing a water softener is not an option (e.g., for apartment dwellers). As noted, the contrast between hard and soft water will be particularly evident with shaving soap, so try that.
Get a gallon of distilled (aka “purified” water) at the drugstore, where it sells for about $1/gallon for use in steam irons, steamers, vaporizers, and the like. Heat 1 cup of it to around 115ºF-120ºF (46ºC-49ºC).
For boar brushes: Pour enough hot water in a coffee cup to soak the boar brush while you shower. The remaining water in the cup can be used to rinse the razor’s head of lather during the shave. For other brushes, just pour about 1″ of water in the cup for razor rinsing and simply dip the brush into the reserved water to wet it throughly.
Load the brush with soap by brushing the tips briskly and with a certain firmness over the puck, and continue even after you first start to see some lather: you want to get enough soap into the brush for a thick and creamy lather.
You may need to add just a little water—on the order of 1 teaspoon—to bring the lather along, but since it’s distilled water, the lather will be abundant. Work up the lather on your beard or in a bowl, and shave.
I begin my shaves by washing my beard with Musgo Real Lime Glyce Soap (MR GLO), but that takes almost no water at all: a spoonful-amount to wet the bar so I can get some soap on my hands, use that to wash my beard, and “rinse” with another spoonful-amount. This is not a thorough rinse: the residual MR GLO adds lubricity.
You can rinse your hands under the hot-water tap: no need to use distilled water on your hands. For the same reason, a warming scuttle, if you use one, does not need distilled water: again, just use water from the hot-water tap.
The “rinse” following the first and second passes can, like the “rinse” of MR GLO, is more a matter of using a spoonful-amount of water to wet your face than it is a true rinse. Only the final rinse needs to be thorough.
Use any remaining distilled water to rinse the brush as best you can, then finish the rinse under the hot water tap until the brush is cleaned of soap, followed by a final rinse in cold tap water.
With a little practice, you can get a good shave with 1/2 cup of distilled water, which means a gallon will provide water for 32 shaves: a month, comfortably.
If you try this and discover that you get a much better shave with distilled water, you might consider a water softener: hard water not only makes for bad shaves, it’s also hard on the plumbing and the valves. The best solution is a water softener that provides soft water throughout the house save for outside faucets and the kitchen cold water. (Because softened water is relatively high in sodium, it should not be used for drinking or cooking.) The best softeners regenerate based on volume rather than time, which automatically accommodates periods of low usage (vacations, for example) and high usage (house guests, for example). A twin-tank softener provides uninterrupted soft water even when the unit is regenerating.
UPDATE: Note that bottled drinking water is hard water, which tastes better than distilled water. A typical Brita filter removes only particulates, but the Brita Maxtra filter, as pointed out by Joachim Moeller in comments, will indeed soften water. However, a gallon of distilled water is only $1, so you might want to try that before buying a Brita Maxtra to be sure that any lathering problems are indeed due to hard water.